On the Theory of Polynomial Ideals and Resultants

In what follows the question is the general problem of elimination theory, namely to determine the common zeros of all polynomials of an ideal of polynomials; or, what is identical with this, the zeros of any finite number of polynomials that form a basis of the ideal. At the same time a conceptual interpretation of the multiplicities that occur will emerge. The coefficients of the polynomials may be assigned to any field; the zeros then belong to the algebraically closed field derived from it. Moreover, the ideal may be assumed, without restriction of generality, to be transformed (§ 3). Then the essential result is the following:

To every ideal \(\mathfrak m\) of polynomials one can assign a resultant form: \[R_{\mathfrak m}=R^{(1)}(x_1,\ldots,x_n)\cdots R^{(i)}(x_i,\ldots,x_n)\cdots R^{(n)}(x_n)\equiv 0(\mathfrak m),\] in such a way that \(R_{\mathfrak m}\) vanishes for all zeros of \(\mathfrak m\) and only for these. If \(\mathfrak n\) is a divisor of \(\mathfrak m\), and if the resultant forms \(R_{\mathfrak n}\) and \(R_{\mathfrak m}\) agree, then the ideals \(\mathfrak n\) and \(\mathfrak m\) also agree.¹

The second part of the main theorem shows that the resultant form supplies the zeros in characteristic multiplicity. This second part is analogous to the fact that two ideals of an algebraic number field, one of which is a divisor of the other, agree if and only if their norms agree; the inner reason for the two theorems is also the same.

Namely, \(\mathfrak m\) may be conceived as a module of linear forms \(\mathfrak M_{i-1}\), by regarding each polynomial as a linear form in the power products of \(x_1,\ldots,x_{i-1}\) and adjoining \(x_{i+1},\ldots,x_n\) to the coefficient domain. The resultant factor \(R^{(i)}\) then becomes equal to the norm of the corresponding fundamental module (§ 1) \(\mathfrak G_{i-1}\) with respect to \(\mathfrak M_{i-1}\); this also immediately gives an interpretation of the multiplicity – product of degree and exponent of a factor of \(R^{(i)}\) – by the number of linearly independent residue classes², a fact that until now was known only in the special case in which the resultant can be defined for indeterminate coefficients³.

For the derivation of these results, §§ 1 and 2 give theorems on modules of linear forms whose coefficients are rational integers or polynomials in one indeterminate. The connection with the ideals of polynomials introduced in § 3 is supplied by the isomorphism theorem of § 4. Section 5 gives the above-mentioned second part of the main theorem; § 7 gives the theorem on the zeros, preceded in § 6 by a general elimination theory. There, at the same time, when new indeterminates are introduced, the decomposition of the resolvent into linear forms in these indeterminates is proved; and thus, for the elimination given here, an unobjectionable proof is supplied for an assertion of Kronecker⁴.

§ 1. Modules of Linear Forms.

By integral quantities \(a,b,c,\ldots\) in the first two sections we shall mean rational integers or polynomials in one indeterminate with coefficients from an arbitrary, abstractly defined field \(P\); by linear forms \(a(\xi),b(\xi),\ldots\) we shall mean such forms in finitely many indeterminates \(\xi_1,\ldots,\xi_k\) with integral quantities as coefficients.

A module \(\mathfrak A\) of linear forms is defined by the conditions: together with \(a(\xi)\) and \(b(\xi)\), \(\mathfrak A\) contains the difference \(a(\xi)-b(\xi)\); together with \(a(\xi)\) it contains \(c\cdot a(\xi)\), where \(c\) denotes an arbitrary integral quantity.

By the rank of \(\mathfrak A\) we mean, as usual, the maximal number of linearly independent linear forms from \(\mathfrak A\); by the \(\lambda\)-th determinantal divisor \(d_\lambda\) of \(\mathfrak A\), the greatest common divisor of all determinants of order \(\lambda\) from \(\mathfrak A\) (that is, all determinants of order \(\lambda\) formed from the coefficient matrix of any \(\lambda\) linear forms from \(\mathfrak A\)). If \(\rho\) is the rank of \(\mathfrak A\), so that \(d_1,\ldots,d_\rho\) are the non-zero determinantal divisors, then the elementary divisors of \(\mathfrak A\) are defined by \(e_1=d_1\), \(e_2=d_2/d_1,\ldots,e_\rho=d_\rho/d_{\rho-1}\), \(e_{\rho+i}=0\). It is known that \(\mathfrak A\) is a finite module having a module basis of exactly \(\rho\) linear forms \(a_1(\xi),\ldots,a_\rho(\xi)\), by which every linear form from \(\mathfrak A\) can be represented linearly, with integral quantities as coefficients: \(\mathfrak A=(a_1(\xi),\ldots,a_\rho(\xi))\). If \(A\) denotes the coefficient matrix of these linear forms, then the determinantal and elementary divisors of \(\mathfrak A\) agree with those of \(A\); hence first it follows that each elementary divisor \(e_i\) divides the following one \(e_{i+1}\). The matrix equation furnished by the elementary-divisor theory, \[P A Q=\begin{pmatrix} e_1&&&0\\ &e_2&&\\ &&\ddots&\\ 0&&&e_\rho \end{pmatrix},\] further shows – if new indeterminates \(\eta_1,\ldots,\eta_k\) are introduced by the unimodular transformation \(\xi=Q(\eta)\) – that the module equation holds: \[\begin{equation} \mathfrak A=(e_1\eta_1,e_2\eta_2,\ldots,e_\rho\eta_\rho). \tag{1} \end{equation}\]

The concept most essential for what follows is that of the fundamental module⁵, according to

Definition I. A module \(\mathfrak G\) is called a fundamental module if it has no proper divisor of the same rank⁶.

Thus \(\mathfrak G\) is a fundamental module if and only if from \[\begin{equation} c\cdot g(\xi)\equiv 0(\mathfrak G);\quad c\ne0 \quad\text{there always follows:}\quad g(\xi)\equiv 0(\mathfrak G). \tag{2} \end{equation}\]

The connection between an arbitrary module and a fundamental module is given by

Theorem I. Every module \(\mathfrak A\) is divisible by one and only one fundamental module \(\mathfrak G\) of the same rank; \(\mathfrak G\) is defined as the smallest divisor of \(\mathfrak A\) having the same rank, and is represented by \[\begin{equation} \mathfrak G=(\eta_1,\ldots,\eta_\rho); \tag{3} \end{equation}\] \(\mathfrak G\) shall be called the fundamental module of \(\mathfrak A\).

The condition that \(\mathfrak G\) is to be the smallest divisor of the same rank is expressed as follows: \(\mathfrak G\) contains all and only the linear forms \(g(\xi)\) that satisfy a relation \[\begin{equation} b\cdot g(\xi)\equiv 0(\mathfrak A);\quad b\ne0. \tag{4} \end{equation}\] It follows first that \(\mathfrak G\) is a module, since together with \[b_1g_1(\xi)\equiv 0(\mathfrak A);\quad b_1\ne0; \qquad b_2g_2(\xi)\equiv 0(\mathfrak A);\quad b_2\ne0\] also \[b_1b_2\bigl(g_1(\xi)-g_2(\xi)\bigr)\equiv 0(\mathfrak A);\quad b_1b_2\ne0\] is fulfilled; and of course also \(b_1\cdot c g_1(\xi)\equiv 0(\mathfrak A)\). But \(\mathfrak G\) is also a fundamental module; for from \[c\cdot g(\xi)\equiv 0(\mathfrak G);\quad c\ne0\] there follows, by (4), \[b c g(\xi)\equiv 0(\mathfrak A);\quad bc\ne0,\] and hence again by (4) also \(g(\xi)\equiv 0(\mathfrak G)\).

There is moreover no fundamental module \(\mathfrak G_1\) different from the smallest divisor \(\mathfrak G\) of the same rank that satisfies the conditions. For \(\mathfrak G_1\) would have to be divisible by \(\mathfrak G\), but as a fundamental module it has no proper divisor of the same rank, and hence is identical with \(\mathfrak G\). Finally, the module represented by the right hand side of (3) is a fundamental module, since every proper divisor must contain a further indeterminate \(\eta\), and so has higher rank. Thus (3) gives the uniquely defined fundamental module of \(\mathfrak A\), and Theorem I is proved in all its parts.

A further concept essential for what follows is Dedekind’s concept of the quotient of two modules⁷:

Definition II. The quotient \(\mathfrak c=\mathfrak A/\mathfrak B\) of two modules of linear forms is defined as the totality of the integral quantities \(c\) satisfying the condition \[c\cdot\mathfrak B\equiv 0(\mathfrak A).\] \(\mathfrak c\) is an ideal of integral quantities, hence a principal ideal. Correspondingly, the quotient \(\mathfrak C=\mathfrak A/\mathfrak b\) of a module \(\mathfrak A\) of linear forms by an ideal \(\mathfrak b\) of integral quantities is defined as the totality of the linear forms \(c(\xi)\) satisfying the condition \[c(\xi)\cdot\mathfrak b\equiv 0(\mathfrak A).\] \(\mathfrak C\) is again a module of linear forms, and indeed, as soon as \(\mathfrak b\) is different from the zero ideal, a module of the same rank as \(\mathfrak A\).

One need only remark that by definition it is clear that the module property belongs to the quotient in each case; systems of integral quantities with the module property, however, are ideals.

The quotient concept leads to a connection between the fundamental module and the highest elementary divisor, namely:

Theorem II. Between the fundamental module and the highest elementary divisor of \(\mathfrak A\) there is the reciprocal relation \[\begin{equation} (e_\rho)=\mathfrak A/\mathfrak G;\qquad \mathfrak G=\mathfrak A/(e_\rho), \tag{5} \end{equation}\] where \((e_\rho)\) denotes the principal ideal derived from \(e_\rho\).

For since every elementary divisor divides \(e_\rho\), from (1) and (3) we have \[\begin{equation} e_\rho\mathfrak G\equiv 0(\mathfrak A)\quad \text{and consequently}\quad (e_\rho)\cdot\mathfrak G\equiv 0(\mathfrak A); \tag{6} \end{equation}\] hence \((e_\rho)\) is divisible by the quotient \(\mathfrak A/\mathfrak G\). But the converse also holds; for from \[b\mathfrak G\equiv 0(\mathfrak A) \quad\text{there follows}\quad b\eta_\rho\equiv 0(\mathfrak A) \quad\text{and hence}\quad b\equiv 0((e_\rho)),\] which proves the first formula (5)⁸. Formula (6) further shows that \(\mathfrak G\) is divisible by the quotient \(\mathfrak A/(e_\rho)\); this quotient is a divisor of \(\mathfrak A\) of the same rank, hence, by the definition of the fundamental module of \(\mathfrak A\), conversely divisible by \(\mathfrak G\). Thus the second formula (5) is also proved.

If \(d\) denotes any determinant of order \(\rho\) from \(\mathfrak A\) which does not vanish identically, then \(d\) is divisible by the \(\rho\)-th determinantal divisor \(d_\rho\), and hence by \(e_\rho\); consequently \(d\mathfrak G\equiv 0(\mathfrak A)\), and by the preceding conclusion \(\mathfrak G=\mathfrak A/(d)\). For what follows, however, it is essential that this latter quotient representation for \(\mathfrak G\) can be proved without going back to the elementary divisors, and hence has a more general validity, according to

Theorem III. If by integral quantities one understands polynomials in several indeterminates⁹, then the quotient representation still holds \[\begin{equation} \mathfrak G=\mathfrak A/(d), \tag{7} \end{equation}\] where \(d\) denotes any determinant of order \(\rho\) from \(\mathfrak A\) which does not vanish identically.

First it is clear that the notions of rank, fundamental module and module quotient – which now is just no longer a principal ideal – are preserved under this extension, and so is Theorem I, with the exception of the basis representation.

Let now \[a_1(\xi)=a_{11}\xi_1+\cdots+a_{1k}\xi_k,\ldots, a_\rho(\xi)=a_{\rho1}\xi_1+\cdots+a_{\rho k}\xi_k\] be \(\rho\) linearly independent linear forms from \(\mathfrak A\), which in general will not form a module basis. Let the indeterminates \(\xi\) be so designated that \[d=|a_{ij}|\ne0;\qquad i,j=1,2,\ldots,\rho.\] It follows that \(\rho\) linear forms of special shape belong to \(\mathfrak A\): \[\begin{equation} d\xi_\lambda+b_{\lambda,\rho+1}\xi_{\rho+1}+\cdots+b_{\lambda,k}\xi_k\equiv 0(\mathfrak A) \quad(\lambda=1,2,\ldots,\rho). \tag{8} \end{equation}\] But \(\mathfrak A\) can contain no linear form depending only on \(\xi_{\rho+1},\ldots,\xi_k\), say \(c_{\rho+1}\xi_{\rho+1}+\cdots+c_k\xi_k\), since otherwise at least one determinant of order \(\rho+1\), namely \(d\cdot c_\alpha\), would be non-zero.

Now the quotient \(\mathfrak A/(d)\), as a divisor of \(\mathfrak A\) of the same rank, is divisible by \(\mathfrak G\); but the converse also holds. For every \(g(\xi)\equiv 0(\mathfrak G)\) satisfies, by definition, \[c\cdot g(\xi)\equiv 0(\mathfrak A);\quad c\ne0 \quad\text{and consequently}\quad d\cdot c\cdot g(\xi)\equiv 0(\mathfrak A).\] By (8), however, \[d\cdot g(\xi)=g_{\rho+1}\xi_{\rho+1}+\cdots+g_k\xi_k\equiv 0(\mathfrak A),\] and consequently \[c g_{\rho+1}\xi_{\rho+1}+\cdots+c g_k\xi_k\equiv 0(\mathfrak A).\] By what has just been observed this implies \(c\cdot g_{\rho+1}=0,\ldots,c\cdot g_k=0\), and because \(c\ne0\), also \(g_{\rho+1}=0,\ldots,g_k=0\); hence \(d\cdot g(\xi)\equiv 0(\mathfrak A)\), proving (7).

It should be noted that \(d\cdot g(\xi)\equiv 0(\mathfrak A)\) also follows directly from the fact that the two modules \((a_1(\xi),\ldots,a_\rho(\xi))\) and \((g(\xi),a_1(\xi),\ldots,a_\rho(\xi))\) have the same rank \(\rho\), so that – putting \(g(\xi)=c_1\xi_1+\cdots+c_k\xi_k\) – the determinant of order \(\rho+1\) \[\begin{vmatrix} a_1(\xi)&a_{11}&\cdots&a_{1\rho}\\ \vdots&\vdots&&\vdots\\ a_\rho(\xi)&a_{\rho1}&\cdots&a_{\rho\rho}\\ g(\xi)&c_1&\cdots&c_\rho \end{vmatrix}\] vanishes. The proof given above, however, recurs in § 4 following formula (30).

§ 2. Decomposition Theorems for Norms and Modules.

We now return to modules of linear forms with respect to rational integers or polynomials in one indeterminate with coefficients from \(P\).

Definition III. If, as usual, all linear forms in \(\xi\) that are congruent to one another modulo \(\mathfrak A\) are collected into one residue class, then the symbol \(\mathfrak G\mid\mathfrak A\) shall denote the system of residue classes of \(\mathfrak G\) modulo \(\mathfrak A\), that is, the system of all residue classes modulo \(\mathfrak A\) that consist of elements of \(\mathfrak G\). The norm of \(\mathfrak G\) with respect to \(\mathfrak A\) – in symbols \(N(\mathfrak G\mid\mathfrak A)\) – is defined as the determinant of the transition substitution from \(\mathfrak G\) to \(\mathfrak A\), that is, as the determinant of the substitution which expresses any linearly independent module basis of \(\mathfrak A\) by such a basis of the fundamental module \(\mathfrak G\) of \(\mathfrak A\).

From (1) and (3), respectively from the remark to Theorem II, one obtains: \[\begin{equation} N(\mathfrak G\mid\mathfrak A)=e_1e_2\cdots e_\rho=d_\rho; \qquad \mathfrak G=\mathfrak A/\bigl(N(\mathfrak G\mid\mathfrak A)\bigr). \tag{9} \end{equation}\]

From (1), (3) and (9) it follows in the familiar way that, in the case of rational integers, \(N(\mathfrak G\mid\mathfrak A)\) represents the number of residue classes of \(\mathfrak G\) modulo \(\mathfrak A\), whereas in the case of polynomials in one indeterminate – where the number of residue classes is in general infinite – the degree of \(N(\mathfrak G\mid\mathfrak A)\) becomes equal to the number of residue classes linearly independent with respect to \(P\). If \(\mathfrak B\) is a divisor of \(\mathfrak A\) of the same rank, then, together with a representative of any such residue class, \(\mathfrak B\) also contains all elements of the residue class; hence \(\mathfrak B\) arises from \(\mathfrak A\) by adjoining finitely many residue classes, or their linear combinations. It follows at once that:

Theorem IV. If \(\mathfrak B\) is a divisor of \(\mathfrak A\) of the same rank, so that the fundamental modules agree, and if moreover \(N(\mathfrak G\mid\mathfrak B)=N(\mathfrak G\mid\mathfrak A)\), then also \(\mathfrak B=\mathfrak A\).

On the connection between the decomposition of norms and modules one has

Theorem V. Let \[\begin{equation} N(\mathfrak G\mid\mathfrak A)=r\cdot s,\qquad (r,s)=1; \tag{10} \end{equation}\] then there exists one and only one module \(\mathfrak R\), and likewise one and only one module \(\mathfrak S\), both divisors of \(\mathfrak A\) of the same rank, such that \[r=N(\mathfrak G\mid\mathfrak R),\qquad s=N(\mathfrak G\mid\mathfrak S)\] holds. \(\mathfrak R\) and \(\mathfrak S\) are defined by \[\mathfrak R=\mathfrak A/(s);\qquad \mathfrak S=\mathfrak A/(r),\] and \(\mathfrak A\) is equal to the least common multiple of \(\mathfrak R\) and \(\mathfrak S\).¹⁰ If one sets \(r_\rho=(r,e_\rho)\), \(s_\rho=(s,e_\rho)\), then the reciprocities hold \[\begin{equation} (s_\rho)=\mathfrak A/\mathfrak R;\quad \mathfrak R=\mathfrak A/(s_\rho); \qquad (r_\rho)=\mathfrak A/\mathfrak S;\quad \mathfrak S=\mathfrak A/(r_\rho). \tag{11} \end{equation}\]

Indeed, setting generally \(r_i=(r,e_i)\), \(s_i=(s,e_i)\), one obtains from (9) and (10): \[(r_i,s_i)=1;\qquad e_i=r_is_i;\qquad r=r_1\cdots r_\rho;\qquad s=s_1\cdots s_\rho,\] and each \(r_i\) respectively \(s_i\) divides the following \(r_{i+1}\) respectively \(s_{i+1}\). Thus, if one sets \[\mathfrak R=(r_1\eta_1,\ldots,r_\rho\eta_\rho);\qquad \mathfrak S=(s_1\eta_1,\ldots,s_\rho\eta_\rho),\] then \(\mathfrak R\) and \(\mathfrak S\) are divisors of \(\mathfrak A\) of the same rank; owing to the relative primeness of \(r_i\) and \(s_i\), \(\mathfrak A\) becomes the least common multiple of \(\mathfrak R\) and \(\mathfrak S\), and moreover \[N(\mathfrak G\mid\mathfrak R)=r_1\cdots r_\rho=r;\qquad N(\mathfrak G\mid\mathfrak S)=s_1\cdots s_\rho=s.\] Furthermore \[s_\rho\mathfrak R\equiv 0(\mathfrak A);\qquad r_\rho\mathfrak S\equiv 0(\mathfrak A);\] and from \(c\mathfrak R\equiv 0(\mathfrak A)\) follows \(cr_\rho\eta_\rho\equiv 0(\mathfrak A)\), hence \(c\equiv 0((s_\rho))\), proving \((s_\rho)=\mathfrak A/\mathfrak R\), and similarly \((r_\rho)=\mathfrak A/\mathfrak S\). From \[b(\eta)=b_1\eta_1+\cdots+b_\rho\eta_\rho\equiv 0(\mathfrak A/(s_\rho))\] it further follows, because all \(r_i\) are relatively prime to \(s_\rho\), that \(b_i\equiv 0((r_i))\); hence \(\mathfrak R=\mathfrak A/(s_\rho)\) and similarly \(\mathfrak S=\mathfrak A/(r_\rho)\). Thus the reciprocities (11) are proved; in the special case \(r=1\) they pass into (5).

It remains to show the unique determination of \(\mathfrak R\) and \(\mathfrak S\). Let \(\overline{\mathfrak R}\) and \(\overline{\mathfrak S}\) be modules satisfying the conditions of Theorem V, and let \(\bar r_i\) and \(\bar s_i\) be their elementary divisors. Since \(\bar r_i\) and \(\bar s_i\) are divisors of \(e_i\), it follows, because \[r=\bar r_1\cdots\bar r_\rho, \qquad s=\bar s_1\cdots\bar s_\rho, \qquad (\bar r_i,\bar s_i)=1,\] that also \[e_i=\bar r_i\bar s_i, \qquad \bar r_i=(r,e_i)=r_i, \qquad \bar s_i=(s,e_i)=s_i.\] Thus \(\overline{\mathfrak R}\) and \(\overline{\mathfrak S}\) agree with \(\mathfrak R\) and \(\mathfrak S\) in elementary divisors and in fundamental module. If therefore, by a unimodular substitution \(\eta=Q(\xi)\), new indeterminates \(\xi\) are introduced, then \[\overline{\mathfrak R}=(r_1\xi_1,\ldots,r_\rho\xi_\rho);\] \[\mathfrak A=\bigl(r_1s_1(q_{11}\xi_1+\cdots+q_{1\rho}\xi_\rho),\ldots, r_\rho s_\rho(q_{\rho1}\xi_1+\cdots+q_{\rho\rho}\xi_\rho)\bigr).\] From \(\mathfrak A\equiv 0(\mathfrak R)\) one thus obtains \[r_i s_i(q_{i1}\xi_1+\cdots+q_{i\rho}\xi_\rho)\equiv 0(\mathfrak R);\] therefore \(r_is_iq_{ij}\equiv 0((r_i))\). Since \((r,s)=1\), this gives \(r_iq_{ij}\equiv 0((r_i))\), or \(\overline{\mathfrak R}\equiv 0(\mathfrak R)\). But since \(N(\mathfrak G\mid\mathfrak R)=N(\mathfrak G\mid\overline{\mathfrak R})\), Theorem IV gives \(\mathfrak R=\overline{\mathfrak R}\), and correspondingly \(\mathfrak S=\overline{\mathfrak S}\). Theorem V is thereby proved in all its parts.

§ 3. Ideals of Polynomials.

Let \(\overline{\mathfrak m}\) be an ideal of polynomials in the \(n\) indeterminates \(y_1,\ldots,y_n\), with coefficients from an arbitrary, abstractly defined field \(P\); that is, together with \(\Phi_1(y)\) and \(\Phi_2(y)\), \(\overline{\mathfrak m}\) also contains the difference \(\Phi_1(y)-\Phi_2(y)\), and together with \(\Phi(y)\) it also contains \(A(y)\cdot\Phi(y)\), where \(A(y)\) denotes an arbitrary polynomial with coefficients from \(P\)¹¹.

Let the \(y\) be subjected to a transformation with indeterminates as coefficients, \[\begin{equation} y=U(x);\quad \text{for instance}\quad \begin{cases} y_1=x_1,\\ y_2=u_{21}x_1+x_2,\\ \cdots\\ y_n=u_{n1}x_1+\cdots+u_{n,n-1}x_{n-1}+x_n, \end{cases} \tag{12} \end{equation}\] and adjoin the indeterminates \(u_{\mu\nu}\) to the coefficient domain of the polynomials, which thereby passes into a field \(P(u)\). By this adjunction let \(\overline{\mathfrak m}\) pass into \(\overline{\mathfrak m}_{(u)}\); hence \(\overline{\mathfrak m}_{(u)}\) contains, besides the polynomials \(\Phi_\lambda(y)\) from \(\overline{\mathfrak m}\), all linear combinations \(\sum \alpha_\lambda(u)\Phi_\lambda(y)\) of finitely many \(\Phi_\lambda\), where \(\alpha_\lambda(u)\) denotes rational functions of the \(u_{\mu\nu}\) with coefficients from \(P\). By multiplication with a suitable polynomial in \(u\), the \(\alpha_\lambda(u)\) may, without restriction of generality, be assumed to be power products in \(u\). Thus one has \[\begin{equation} \sum \alpha_\lambda(u)\Phi_\lambda(y)\equiv 0(\overline{\mathfrak m}_{(u)}); \qquad \Phi_\lambda(y)\equiv 0(\overline{\mathfrak m})\quad\text{and conversely}. \tag{13} \end{equation}\] By means of (12), \(\overline{\mathfrak m}_{(u)}\) passes into an ideal \(\mathfrak m\) of polynomials in \(x_1,\ldots,x_n\) with coefficients from \(P(u)\): \[\begin{equation} \overline{\mathfrak m}_{(u)}=\mathfrak m\quad \text{by means of}\quad y=U(x). \tag{14} \end{equation}\] The combination of the relations (14) and (13) says the following: \[\begin{equation} \text{From } F(x)\equiv 0(\mathfrak m);\quad F(x)=\Phi(y)=\sum \alpha_\lambda(u)\Phi_\lambda(y)=\sum \alpha_\lambda(u)F_\lambda(x) \text{ follows } F_\lambda(x)\equiv 0(\mathfrak m); \tag{15} \end{equation}\] whereas (14) and (15) conversely yield (13) again.

Definition IV. Ideals \(\mathfrak m\) of polynomials in \(x_1,\ldots,x_n\) with coefficients from \(P(u)\) which satisfy the relations (15), and hence arise from \(\overline{\mathfrak m}_{(u)}\) by means of \(y=U(x)\), shall be called transformed ideals.

Transformed ideals are distinguished by the existence of regular polynomials; a polynomial of degree \(r\) is called regular with respect to \(x_i\) if it contains the term \(x_i^r\) with non-zero coefficient. One sees that the polynomials \(F_i(x)\) are regular with respect to \(x_1\). In what follows the main point will be to regard these transformed ideals as modules of linear forms in certain power products of the \(x\). For this the concept of the fundamental ideal must be fixed; later it will pass into the fundamental module. We first give a notation that will be maintained throughout from now on:

Notation. By \(F^{(i)},f^{(i)},a^{(i)},\ldots\) we shall always mean polynomials in the \(x\) that are free of \(x_1,\ldots,x_{i-1}\).

Definition V. To every ideal \(\mathfrak m\) there are defined \(n\) fundamental ideals \(\mathfrak g_0,\mathfrak g_1,\ldots,\mathfrak g_{n-1}\) by the following stipulation: the fundamental ideal of stage \((i-1)\), \(\mathfrak g_{i-1}\), contains all and only the polynomials \(G(x)\) for which there is a polynomial \(b^{(i)}\) – in general varying with \(G(x)\) – such that \[\begin{equation} b^{(i)}G(x)\equiv 0(\mathfrak m);\qquad b^{(i)}\ne0. \tag{16} \end{equation}\]

That the \(\mathfrak g\) are in fact ideals corresponds exactly to the proof of the module property for \(\mathfrak G\) following formula (4), with (16) playing the analogous role. The ideal \(\mathfrak g_i\) is divisible by \(\mathfrak g_{i-1}\), since every polynomial \(b^{(i+1)}\) is at the same time a \(b^{(i)}\).

For the fundamental ideals one has:

Theorem VI. The fundamental ideals of transformed ideals are transformed ideals¹².

The proof rests on a known theorem of Dedekind–Mertens¹³: let \(a\) and \(b\) be indeterminates, and set \[\sum a_{i_1,\ldots,i_s}z_1^{i_1}\cdots z_s^{i_s}\cdot \sum b_{j_1,\ldots,j_s}z_1^{j_1}\cdots z_s^{j_s} = \sum c_{k_1,\ldots,k_s}z_1^{k_1}\cdots z_s^{k_s},\] \[\sum i\le \nu, \qquad \sum j\le \mu, \qquad \sum k\le \nu+\mu.\] If \(\mathfrak A,\mathfrak B,\mathfrak C\) denote the modules derived respectively from the \(a,b,c\) by means of rational integers, then there exists an exponent \(q\) such that the identity holds: \[\begin{equation} \mathfrak A^q\mathfrak B=\mathfrak A^{q-1}\mathfrak C. \tag{17} \end{equation}\] Here \(\mathfrak A^q\) consists of the power products of \(q\)-th dimension in the \(a\) and their integral linear combinations.

For the proof of Theorem VI, now put \[\begin{equation} \begin{cases} G(x)=\Gamma(y)=\sum \alpha_\lambda(u)\Gamma_\lambda(y)=\sum \alpha_\lambda(u)G_\lambda(x),\\ b^{(i)}(x)=\varphi(y)=\sum \beta_\mu(u)\varphi_\mu(y), \end{cases} \tag{18} \end{equation}\] where \(\alpha_\lambda(u)\) and \(\beta_\mu(u)\) are power products in the \(u\). Then, by (16), (14) and (18), \[\begin{equation} \sum\beta_\mu(u)\varphi_\mu(y)\cdot \sum\alpha_\lambda(u)\Gamma_\lambda(y)\equiv 0(\overline{\mathfrak m}_{(u)}); \qquad \sum\beta_\mu(u)\varphi_\mu(y)\ne0. \tag{19} \end{equation}\] On the left stands the product of two polynomials in \(u\) whose coefficients are the polynomials \(\varphi_\mu(y)\) and \(\Gamma_\lambda(y)\); the coefficients of the product polynomial in \(u\) are, by the right hand side of (19), polynomials from \(\overline{\mathfrak m}\). If, therefore, in (17) one replaces the \(a,b,c\) by these polynomials in \(y\), one obtains \[\left\{\sum\beta_\mu(u)\varphi_\mu(y)\right\}^q\cdot \Gamma_\lambda(y)\equiv 0(\overline{\mathfrak m}_{(u)}),\] which by (18) is equivalent to: \[\begin{equation} b^{(i)}(x)^q\cdot G_\lambda(x)\equiv 0(\mathfrak m); \qquad G_\lambda(x)\equiv 0(\mathfrak g_{i-1}). \tag{20} \end{equation}\] The relations (16), (18) and (20) show that the fundamental ideals \(\mathfrak g_{i-1}\) are indeed transformed ideals.

§ 4. Connection between ideals of polynomials and modules of linear forms.

To regard an ideal \(\mathfrak m\) of polynomials, which is always assumed transformed, as a module \(\mathfrak M_{i-1}\) \((i=1, \ldots,n)\) of linear forms, the individual polynomials \(F(x)\) are to be considered as linear forms in the power products \(\xi_\lambda\) of \(x_1\ldots x_{i-1}\): \[F(x)=\sum a^{(i)}_\lambda\xi_\lambda,\] where, according to the notation of § 3, the \(a^{(i)}_\lambda\) are polynomials in \(x_i\ldots x_n\). From the definition of the ideal it follows at once that \(\mathfrak M_{i-1}\) has the module property with respect to these integral quantities \(a^{(i)},b^{(i)},\ldots\); and since the infinitely many power products \(\xi\) of \(x_1\ldots x_{i-1}\) occur only linearly, by reason of their linear independence they play for \(\mathfrak M_{i-1}\) the role of indeterminates. Each module \(\mathfrak M_{i-1}\) contains infinitely many indeterminates \(\xi\), but in each individual linear form only finitely many indeterminates occur. Likewise each fundamental ideal \(\mathfrak g_{i-1}\) is to be regarded as a module \(\mathfrak G_{i-1}\) of linear forms, where the indeterminates are again the power products of \(x_1\ldots x_{i-1}\). For \(i=1\) there is only one indeterminate \(\xi_0\), corresponding to the unit.

One can now, and this is the basis of everything that follows, restrict oneself, despite these infinitely many indeterminates \(\xi\), to modules of linear forms in finitely many indeterminates, according to the following theorem.

Theorem VII. The residue-class system \(\mathfrak G_{i-1}\mid\mathfrak M_{i-1}\) is isomorphic to a residue-class system \(\mathfrak G^*_{i-1}\mid\mathfrak M^*_{i-1}\), where \(\mathfrak M^*_{i-1}\) is a module in finitely many indeterminates and \(\mathfrak G^*_{i-1}\) is its fundamental module. The module \(\mathfrak M_{i-1}\) has – after adjoining \(x_{i+1}\ldots x_n\) to \(P(u)\) – only finitely many elementary divisors different from \(1\), namely the elementary divisors of \(\mathfrak M^*_{i-1}\) that are different from \(1\).

The elementary divisors of \(\mathfrak M_{i-1}\) are to be defined as in note 8). The isomorphism occurring in Theorem VII rests on the following definition.

Definition V. Two residue-class systems are called isomorphic if they can be put in one-to-one correspondence in such a way that the difference of two classes is assigned to the difference of the corresponding classes, and the product of a class by an integral quantity to the product of the corresponding class by the same integral quantity.

The proof of Theorem VII is obtained by complete induction. For \(i=1\) the theorem is evident: \(\mathfrak M_0\) is a module of linear forms in one indeterminate \(\xi_0\), corresponding to the power product of dimension zero of the \(x\), that is, to the unit; the integral quantities are all polynomials in \(x_1\ldots x_n\). The fundamental ideal \(\mathfrak g_0\) is the unit ideal consisting of all polynomials in \(x_1\ldots x_n\), and hence \(\mathfrak G_0\) is the module \((\xi_0)\), the fundamental module of \(\mathfrak M_0\); thus here \(\mathfrak G^*_0\mid\mathfrak M^*_0\) coincides with \(\mathfrak G_0\mid\mathfrak M_0\). Finally, since \(\mathfrak M_0\) has rank \(1\), it can have at most one elementary divisor different from \(1\).

We first pass from \(i=1\) to \(i=2\), in order to use the insight thereby obtained into the structure of \(\mathfrak G_1\mid\mathfrak M_1\) for the general induction step. For this purpose we first show that for \(\mathfrak G_0\mid\mathfrak M_0\) an \(x_1\)-bounded system of representatives may be chosen; because of the divisibility of \(\mathfrak g_1\) by \(\mathfrak g_0\), this will then lead to the finitely many indeterminates of \(\mathfrak G^*_1\).

As a transformed ideal, \(\mathfrak m\) possesses by § 3 at least one polynomial \(C^{(1)}(x)\) regular in \(x_1\), say of dimension \(k\); hence \(\mathfrak M_0\) contains the linear form \(C^{(1)}\xi_0\). By the regularity of \(C^{(1)}(x)\), every polynomial \(H(x)\) admits a representation \[\begin{equation} H(x)=b^{(2)}_0+b^{(2)}_1x_1+\cdots+b^{(2)}_{k-1}x_1^{k-1}\quad(C^{(1)}(x)); \tag{21} \end{equation}\] therefore, since \(C^{(1)}\xi_0\) belongs to \(\mathfrak M_0\), one can indeed choose for \(\mathfrak G_0\mid\mathfrak M_0\) a system of representatives that does not reach the \(k\)-th dimension in \(x_1\).

If now, in particular, for the polynomials \(G(x)\) from \(\mathfrak g_1\) and \(F(x)\) from \(\mathfrak m\) one writes \[\begin{equation} \left\{\begin{aligned} G(x)&=g^{(2)}_0+g^{(2)}_1x_1+\cdots+g^{(2)}_{k-1}x_1^{k-1} &&(C^{(1)}(x)),\\ F(x)&=a^{(2)}_0+a^{(2)}_1x_1+\cdots+a^{(2)}_{k-1}x_1^{k-1} &&(C^{(1)}(x)), \end{aligned}\right. \tag{22} \end{equation}\] and if \(\mathfrak G^*_1\) denotes the system of all linear forms \(g(\xi)=g^{(2)}_0\xi_0+\cdots+g^{(2)}_{k-1}\xi_{k-1}\), and correspondingly \(\mathfrak M^*_1\) the system of the linear forms \(a(\xi)=a^{(2)}_0\xi_0+\cdots+a^{(2)}_{k-1}\xi_{k-1}\), then the ideal property of \(\mathfrak g_1\) and \(\mathfrak m\) shows that \(\mathfrak G^*_1\) and \(\mathfrak M^*_1\) are modules of linear forms in \(\xi_0\ldots\xi_{k-1}\) with respect to the integral quantities \(b^{(2)},c^{(2)},\ldots\). Likewise the principal ideal derived from \(C^{(1)}(x)\) gives a module with respect to these integral quantities, \[\mathfrak G_1=(\xi_0,\xi_1,\ldots,\xi_r,\ldots),\] where \(\xi_r=x_1^rC^{(1)}(x)\). The \(\xi_r\) are linearly independent with respect to these integral quantities \(b^{(2)},c^{(2)},\ldots\), both among themselves and from the finitely many \(\xi\). From the divisibility of \(\mathfrak G_1\) by \(\mathfrak M_1\), and hence by \(\mathfrak G_1\), it follows that \(\mathfrak G^*_1\) is divisible by \(\mathfrak G_1\) and \(\mathfrak M^*_1\) by \(\mathfrak M_1\). Taking (22) into account, one obtains \[\begin{equation} \mathfrak G_1=(\mathfrak G^*_1,\mathfrak G_1),\qquad \mathfrak M_1=(\mathfrak M^*_1,\mathfrak G_1). \tag{23} \end{equation}\]

From (23) and from the linear independence of the \(\xi\) and \(\xi\), Theorem VII for \(i=2\) follows as follows. Because \(\mathfrak G_1\) is divisible by \(\mathfrak M_1\), one first has \(\mathfrak G_1\mid\mathfrak M_1=\mathfrak G^*_1\mid\mathfrak M_1\). To prove the isomorphism with \(\mathfrak G^*_1\mid\mathfrak M^*_1\), let certain linear forms \(g^*(\xi)\) from \(\mathfrak G^*_1\) give a system of representatives of \(\mathfrak G^*_1\mid\mathfrak M^*_1\). Because of the linear independence of the \(\xi\) and \(\xi\), from \(g^*(\xi)\equiv0(\mathfrak M_1)\) it follows also that \(g^*(\xi)\equiv0(\mathfrak M^*_1)\); the \(g^*(\xi)\) are therefore also incongruent modulo \(\mathfrak M_1\), form a system of representatives of \(\mathfrak G_1\mid\mathfrak M_1\), and the correspondence from \(\mathfrak G_1\mid\mathfrak M_1\) to \(\mathfrak G^*_1\mid\mathfrak M^*_1\) mediated by this system of representatives is isomorphic in the sense of Definition V.

In exactly the same way one obtains: from \(b^{(2)}g(\xi)\equiv0(\mathfrak M_1)\), \(b^{(2)}\ne0\), follows \(b^{(2)}g(\xi)\equiv0(\mathfrak M^*_1)\), \(b^{(2)}\ne0\). Since this is satisfied for all \(g(\xi)\) from \(\mathfrak G^*_1\), and only for these – for by (23) \(\mathfrak G^*_1\) consists of all polynomials of the fundamental ideal \(\mathfrak g_1\) that are at the same time linear forms in \(\xi\) – \(\mathfrak G^*_1\) is thereby characterized as the fundamental module of \(\mathfrak M^*_1\).

Adjoining finally \(x_3\ldots x_n\) to \(P(u)\), one gets \[\begin{equation} \left\{\begin{aligned} \mathfrak G^*_1&=(\eta_1,\ldots,\eta_\rho),& \mathfrak M^*_1&=(e_1\eta_1,\ldots,e_\rho\eta_\rho),\\ \mathfrak G_1&=(\eta_\rho,\ldots,\eta_1,\xi_0,\ldots,\xi_r,\ldots),& \mathfrak M_1&=(e_\rho\eta_\rho,\ldots,e_1\eta_1,\xi_0,\ldots,\xi_r,\ldots), \end{aligned}\right. \tag{24} \end{equation}\] and hence \[(e_\rho)=\mathfrak M_1/\mathfrak G_1=\mathfrak M_1/(\mathfrak M_1,\eta_\rho),\qquad (e_{\rho-1})=(\mathfrak M_1,\eta_\rho)/\mathfrak G_1\quad\hbox{etc.}\] Thus, taking note 8) into account, \(e_\rho,e_{\rho-1},\ldots,e_1,1,\ldots,1,\ldots\) are recognized as the elementary divisors of \(\mathfrak M_1\). This proves all parts of Theorem VII for \(i=2\).

It should be noted that (22) and (23) use only that \(C^{(1)}(x)\) is regular in \(x_1\). These formulae, and the arguments attached to them leading to Theorem VII, therefore remain valid if, in place of (12), the special transformation \[\begin{equation} y_1=x_1;\quad y_2=u_{21}x_1+z_2;\quad\ldots;\quad y_n=u_{n1}x_1+z_n \tag{25} \end{equation}\] is taken as the basis; by composition with \[\begin{equation} \left\{\begin{array}{l} z=U_1(x)\quad\hbox{or}\\[2mm] z_2=x_2;\quad z_3=u_{32}x_2+x_3;\quad\ldots;\quad z_n=u_{n2}x_2+\cdots+u_{n,n-1}x_{n-1}+x_n, \end{array}\right. \tag{26} \end{equation}\] this again gives (12). If \(\widetilde{\mathfrak m}_{(u)}\) passes by means of (25) into \(\widetilde{\mathfrak m}\), and if \(\widetilde{\mathfrak g}_1,\widetilde{\mathfrak G}_1,\ldots\) have for \(\widetilde{\mathfrak m}\) the same meaning as \(\mathfrak g_1,\mathfrak G_1,\ldots\) have for \(\mathfrak m\), then \[\begin{equation} \widetilde{\mathfrak G}_1=(\widetilde{\mathfrak G}^{*}_1,\widetilde{\mathfrak G}_1),\qquad \widetilde{\mathfrak M}_1=(\widetilde{\mathfrak M}^{*}_1,\widetilde{\mathfrak G}_1). \tag{27} \end{equation}\]

Here (27) arises from (23) simply by inversion of (26). This is evident for \(\mathfrak m\) and \(C^{(1)}(x)\), and hence also for \(\mathfrak G_1\) and \(\mathfrak M_1\). It is also true for \(\mathfrak g_1\), since \[b^{(2)}(x)G(x)\equiv0(\mathfrak m),\quad b^{(2)}\ne0, \qquad \widetilde b^{(2)}(z)\widetilde G(x,z)\equiv0(\widetilde{\mathfrak{m}}),\quad \widetilde b^{(2)}\ne0\] mutually condition one another through \(z=U_1(x)\). Thus \(\widetilde{\mathfrak{g}}_1=\mathfrak g_1\) by means of (26); hence also \(\widetilde{\mathfrak{G}}_1=\mathfrak G_1\). Consequently, by the definition given through (22), the same holds for \(\mathfrak G_1^*\) and \(\mathfrak M_1^*\) as for \(\widetilde{\mathfrak{G}}_1^*\) and \(\widetilde{\mathfrak{M}}_1^*\), and so for the whole representation (27).

For the general induction step, suppose the following hypotheses hold: \[\begin{equation} \left\{\begin{aligned} \mathfrak G_{i-1}&=(\mathfrak G_{i-1}^*,\mathfrak G_{i-1}),& \mathfrak M_{i-1}&=(\mathfrak M_{i-1}^*,\mathfrak G_{i-1}),\\ \mathfrak G_{i-1}&=(\xi_0,\xi_1,\ldots,\xi_r,\ldots), \end{aligned}\right. \tag{28} \end{equation}\] where \(\mathfrak G^*_{i-1}\) and \(\mathfrak M^*_{i-1}\) are modules, with respect to the integral quantities \(b^{(i)},c^{(i)},\ldots\), of linear forms in finitely many indeterminates \(\xi\), certain power products of \(x_1, \ldots,x_{i-1}\); and where the \(\xi\) are linearly independent, both among themselves and from the \(\xi\), with respect to these integral quantities. A representation corresponding to (28), and the linear independence of the \(\xi\) and \(\xi\), is also to hold when, instead of (12), one uses the special transformation \[\begin{equation} \begin{aligned} y_1&=x_1;\quad \ldots;\quad y_{i-1}=u_{i-1,1}x_1+\cdots+x_{i-1};\\ y_i&=u_{i,1}x_1+\cdots+u_{i,i-1}x_{i-1}+z_i;\quad\ldots;\quad y_n=u_{n,1}x_1+\cdots+u_{n,i-1}x_{i-1}+z_n, \end{aligned} \tag{29} \end{equation}\] which can be composed to give (12) by means of \[z=U_{i-1}(x)\quad\hbox{or}\quad z_i=x_i;\quad z_{i+1}=u_{i+1,i}x_i+x_{i+1};\quad\ldots;\quad z_n=u_{n,i}x_i+\cdots+u_{n,n-1}x_{n-1}+x_n.\] Moreover this special representation is to arise from (28) simply by inversion of \(z=U_{i-1}(x)\).

From the hypotheses (28) and the linear independence of the \(\xi\) and \(\xi\), the isomorphism of \(\mathfrak G_{i-1}\mid\mathfrak M_{i-1}\) with \(\mathfrak G^*_{i-1}\mid\mathfrak M^*_{i-1}\) follows by assignment to the same system of representatives, by exactly the same arguments as those attached above to (23). Likewise it follows that \(\mathfrak G^*_{i-1}\) becomes the fundamental module of \(\mathfrak M^*_{i-1}\), and that \(\mathfrak M_{i-1}\) has only finitely many elementary divisors different from \(1\), namely those elementary divisors of \(\mathfrak M^*_{i-1}\) that are different from \(1\).

To carry out the induction step it remains first to show again that for \(\mathfrak G^*_{i-1}\mid\mathfrak M^*_{i-1}\), and hence also for \(\mathfrak G_{i-1}\mid\mathfrak M_{i-1}\), an \(x_i\)-bounded system of representatives may be chosen. This follows from the quotient representation proved in (7), Theorem III: \[\begin{equation} \mathfrak G^*_{i-1}=\mathfrak M^*_{i-1}/(C^{(i)}), \tag{30} \end{equation}\] where \(C^{(i)}\) means any non-identically vanishing determinant of degree \(\rho\) from \(\mathfrak M^*_{i-1}\), \(\rho\) being the rank of \(\mathfrak M^*_{i-1}\). From the part of the hypotheses referring to the special transformation (29), it follows that \(C^{(i)}\) may always be assumed regular with respect to \(x_i\). For the module \(\widetilde{\mathfrak{M}}^*_{i-1}\) corresponding to \(\mathfrak M^*_{i-1}\) has the same rank; if \(\widetilde C^{(i)}\) denotes any non-identically vanishing determinant of degree \(\rho\) from \(\widetilde{\mathfrak{M}}^*_{i-1}\) which passes by \(z=U_{i-1}(x)\) into a regular \(C^{(i)}\), then by hypothesis \(C^{(i)}\) is a determinant of degree \(\rho\) from \(\mathfrak M^*_{i-1}\).

From the existence of \(C^{(i)}\) it follows, as in (8), with a suitable numbering of the \(\xi\), that \(\rho\) linear forms of the special shape \[L_\lambda(\xi)=C^{(i)}\xi_\lambda+c^{(i)}_{\lambda,1}\xi_{\rho+1}+\cdots+c^{(i)}_{\lambda,s-\rho}\xi_s \qquad(\lambda=1\ldots\rho)\] belong to \(\mathfrak M^*_{i-1}\). Every linear form in \(\xi\) can now be brought into the form \[\begin{equation} H(\xi)=a^{(i)}_1\xi_1+\cdots+a^{(i)}_\rho\xi_\rho+b^{(i)}_1\xi_{\rho+1}+\cdots+b^{(i)}_{s-\rho}\xi_s\quad (L_1(\xi),\ldots,L_\rho(\xi)), \tag{31} \end{equation}\] where \(a^{(i)}_1\ldots a^{(i)}_\rho\) are bounded in \(x_i\) and do not reach the degree \(k\) of \(C^{(i)}\). This representation is unique; for from \[a^{(i)}_1\xi_1+\cdots+a^{(i)}_\rho\xi_\rho+b^{(i)}_1\xi_{\rho+1}+\cdots+b^{(i)}_{s-\rho}\xi_s\equiv0\quad (L_1(\xi),\ldots,L_\rho(\xi))\] it follows, by comparing coefficients in \(\xi_1\ldots\xi_\rho\) and considering the degrees in \(x_i\), that all \(a^{(i)}\) and \(b^{(i)}\) vanish identically.

Let now in particular \[H(\xi)\equiv0(\mathfrak G^*_{i-1});\qquad\hbox{hence}\qquad C^{(i)}H(\xi)\equiv0(\mathfrak M^*_{i-1})\quad\hbox{by (30).}\] As in the proof of Theorem III one obtains \[C^{(i)}H(\xi)-\sum_{\tau=1}^{s-\rho}\{C^{(i)}b^{(i)}_\tau-\sum_\lambda a^{(i)}_\lambda c^{(i)}_{\lambda,\tau}\}\xi_{\rho+\tau} \equiv0(\mathfrak M^*_{i-1}),\] and therefore, since as in Theorem III the coefficients of \(\xi_{\rho+1}\ldots\xi_s\) must vanish, \[C^{(i)}b^{(i)}_\tau=\sum_\lambda a^{(i)}_\lambda c^{(i)}_{\lambda,\tau}\qquad(\tau=1\ldots s-\rho).\] Thus the \(b^{(i)}_\tau\) are also bounded and do not reach the maximal degree of the \(c^{(i)}_{\lambda,\tau}\). The existence of an \(x_i\)-bounded system of representatives for \(\mathfrak G^*_{i-1}\mid\mathfrak M^*_{i-1}\), that is for \(\mathfrak G_{i-1}\mid\mathfrak M_{i-1}\), not reaching a certain fixed degree \(r\), is thereby proved.

Let \(\vartheta_1\ldots\vartheta_t\) denote the finitely many power products \[x_i^\alpha\xi_\lambda\quad(\alpha=0,1,\ldots,k-1;\ \lambda=1,2,\ldots,\rho),\qquad x_i^\beta\xi_{\rho+\tau}\quad(\beta=0,1,\ldots,r-1;\ \tau=1,2,\ldots,s-\rho)\] and arrange the countably infinite expressions \[x_i^\gamma L_\lambda\quad(\gamma=0,1,\ldots\text{ in inf.};\ \lambda=1,2,\ldots,\rho), \qquad x_i^\gamma\xi_\nu\quad(\gamma, \nu=0,1, \ldots\text{ in inf.})\] in a simply infinite sequence \(\omega_0,\omega_1,\ldots,\omega_\mu,\ldots\). Then the \(\vartheta\) and \(\omega\) are linearly independent, both each among themselves and from one another, with respect to the integral quantities \(b^{(i+1)},c^{(i+1)},\ldots\). For from \[\sum c^{(i+1)}_\alpha x_i^\alpha\xi_\lambda+ \sum c^{(i+1)}_\beta x_i^\beta\xi_{\rho+\tau}+ \sum d^{(i+1)}_\gamma x_i^\gamma L_\lambda(\xi)+ \sum e^{(i+1)}_{\gamma\nu}x_i^\gamma\xi_\nu=0,\] each sum extending over finitely many terms, the assumed linear independence of the \(\xi\) and \(\xi\), and of the \(\xi\) among themselves with respect to the integral quantities \(b^{(i)}\), gives the vanishing of all \(e^{(i+1)}_{\gamma\nu}\). The uniqueness of the representation (31) further gives the vanishing of the \(c^{(i+1)}_\alpha\) and \(c^{(i+1)}_\beta\), and finally, because of the linear independence of the \(L_\lambda(\xi)\), the vanishing of the \(d^{(i+1)}_\gamma\).

If the module \((\mathfrak G_{i-1},L_1(\xi),\ldots,L_\rho(\xi))\) is now regarded as a module \(\mathfrak G_i\) with respect to the integral quantities \(b^{(i+1)}\), then \(\mathfrak G_i\) is divisible by \(\mathfrak M_i\), because \(\mathfrak G_{i-1},L_1(\xi),\ldots,L_\rho(\xi)\) are divisible by \(\mathfrak M_{i-1}\), and \[\mathfrak G_i=(\omega_0,\omega_1,\ldots,\omega_\mu,\ldots).\] For each \(H(x)\equiv0(\mathfrak g_{i-1})\), (28), (31), and what has just been proved give \[H(x)=b^{(i+1)}_1\vartheta_1+\cdots+b^{(i+1)}_t\vartheta_t\quad(\mathfrak G_i)\] as a module equation with respect to the integral quantities \(b^{(i+1)},c^{(i+1)}\). Since \(\mathfrak g_i\) is divisible by \(\mathfrak g_{i-1}\) and \(\mathfrak m\) is divisible by \(\mathfrak g_i\), if one now sets, in particular, for every \(G(x)\) from \(\mathfrak G_i\) and every \(F(x)\) from \(\mathfrak M_i\), \[G(x)=g^{(i+1)}_1\vartheta_1+\cdots+g^{(i+1)}_t\vartheta_t\quad(\mathfrak G_i), \qquad F(x)=a^{(i+1)}_1\vartheta_1+\cdots+a^{(i+1)}_t\vartheta_t\quad(\mathfrak G_i),\] and denotes by \(\mathfrak G_i^*\) the module consisting of all \(g(\vartheta)\), and by \(\mathfrak M_i^*\) the module consisting of all \(a(\vartheta)\), then exactly the arguments leading to (23) give \[\begin{equation} \mathfrak G_i=(\mathfrak G_i^*,\mathfrak G_i),\qquad \mathfrak M_i=(\mathfrak M_i^*,\mathfrak G_i),\qquad \mathfrak G_i=(\omega_0,\omega_1,\ldots,\omega_\mu,\ldots). \tag{32} \end{equation}\] In deriving (32), again only the regularity of \(C^{(i)}\) in \(x_i\) was used, so the whole argument remains valid if the special transformation (29) is taken with the index increased by one. The same considerations as those attached to (27) then show that this special representation arises from (32) simply by the inversion of \(z=U_i(x)\).

Thus all hypotheses (28), etc., of the general induction step have been proved for the next index; and since, as shown there, Theorem VII follows directly from these hypotheses, Theorem VII is proved in general. At the same time it has been shown that the arbitrariness in \((\mathfrak G^*_{i-1},\mathfrak M^*_{i-1})\) caused by the arbitrary choice of \(C^{(i)}\) is again removed by the isomorphism of the residue-class system \(\mathfrak G^*_{i-1}\mid\mathfrak M^*_{i-1}\).

In particular, if the field \(P\) used in § 3 as coefficient domain has characteristic zero – that is, if the prime field contained in \(P\) and derived from the unit is of the type of the rational numbers – then the indeterminates \(u_{\mu\nu}\) can be specialized to quantities \(\bar u_{\mu\nu}\) from \(P\) in such a way that for \(i=1\ldots n\) each \(C^{(i)}\) remains regular in \(x_i\). The above arguments show that Theorem VII remains valid under this specialization as well.¹⁴ The fundamental module and elementary divisors need not, however, arise necessarily from the general case by the specialization \(u_{\mu\nu}=\bar u_{\mu\nu}\).¹⁵

§ 5. Resultant form and elementary-divisor form of an ideal of polynomials.

Let \(x_{i+1}\ldots x_n\) be adjoined to \(P(u)\), so that \(\mathfrak G_{i-1}\) and \(\mathfrak M_{i-1}\), and therefore also \(\mathfrak G^*_{i-1}\) and \(\mathfrak M^*_{i-1}\), pass into modules of linear forms in which the integral quantities can be regarded as polynomials in one indeterminate \(x_i\). By Theorem VII, in this case the elementary divisors different from \(1\) of \(\mathfrak M_{i-1}\), together with at most finitely many elementary divisors equal to \(1\), agree with those of \(\mathfrak M^*_{i-1}\). The product of these elementary divisors, the \(\rho\)-th determinantal divisor of \(\mathfrak M^*_{i-1}\), was equal to the determinant of the transition substitution from \(\mathfrak G^*_{i-1}\) to \(\mathfrak M^*_{i-1}\), hence to \(N(\mathfrak G^*_{i-1}\mid\mathfrak M^*_{i-1})\) by Definition III. Formula (24), or the analogous formula following from (28) for general \(i\), shows that the product of these elementary divisors is also equal to the determinant of the transition substitution from \(\mathfrak G_{i-1}\) to \(\mathfrak M_{i-1}\), and is therefore to be denoted by \(N(\mathfrak G_{i-1}\mid\mathfrak M_{i-1})\). Thus \[N(\mathfrak G_{i-1}\mid\mathfrak M_{i-1})=N(\mathfrak G^*_{i-1}\mid\mathfrak M^*_{i-1})=R^{(i)}(x),\] where the polynomial \(R^{(i)}(x)\), that is, the greatest common divisor, in the polynomial sense, of all \(\rho\)-rowed determinants from \(\mathfrak M^*_{i-1}\), may be assumed integral and primitive in \(x_{i+1}\ldots x_n\), and consequently, as a divisor of \(C^{(i)}\), becomes regular in \(x_i\). Likewise the highest elementary divisor \(E^{(i)}(x)\) of \(\mathfrak M_{i-1}\), respectively of \(\mathfrak M^*_{i-1}\), may be assumed integral and primitive in \(x_{i+1}\ldots x_n\) and hence regular in \(x_i\). This leads to the following definition.

Definition VI. The polynomial \(R^{(i)}(x)\) is called the resultant of the \(i\)-th stage of \(\mathfrak m\), and \(E^{(i)}(x)\) the elementary divisor of the \(i\)-th stage. Thus the resultant of the \(i\)-th stage is the norm of the fundamental ideal \(\mathfrak g_{i-1}\) with respect to \(\mathfrak m\), interpreted as the norm of the module \(\mathfrak G_{i-1}\) with respect to \(\mathfrak M_{i-1}\), where \(x_{i+1}\ldots x_n\) are adjoined to \(P(u)\); likewise \(E^{(i)}(x)\), under the same adjunction, becomes equal to the quotient \(\mathfrak M_{i-1}/\mathfrak G_{i-1}\). The resultant form and elementary-divisor form of \(\mathfrak m\) are the products \[R_\mathfrak m=R^{(1)}R^{(2)}\cdots R^{(n)},\qquad E_\mathfrak m=E^{(1)}E^{(2)}\cdots E^{(n)}.\]

For resultant form and elementary-divisor form one has:

Theorem VIII. The resultant form is divisible by the elementary form; conversely, a power of \(E_\mathfrak m\) is divisible by \(R_\mathfrak m\). The forms \(E_\mathfrak m\) and \(R_\mathfrak m\) are divisible by \(\mathfrak m\); more generally, \(E^{(i)}\cdots E^{(n)}\mathfrak g_{i-1}\equiv0(\mathfrak m)\) and consequently \(R^{(i)}\cdots R^{(n)}\mathfrak g_{i-1}\equiv0(\mathfrak m)\).

Indeed, since the norm is divisible by the highest elementary divisor and conversely a power of this highest elementary divisor is divisible by the norm, this divisibility follows for \(R^{(i)}(x)\) and \(E^{(i)}(x)\) after adjoining \(x_{i+1}\ldots x_n\). But \(R^{(i)}(x)\) and \(E^{(i)}(x)\) are assumed primitive polynomials with respect to \(x_{i+1}\ldots x_n\); hence the divisibility holds with respect to all indeterminates \(x_i,x_{i+1},\ldots,x_n\). Thus \(R_\mathfrak m\) is divisible by \(E_\mathfrak m\), and some power of \(E_\mathfrak m\) by \(R_\mathfrak m\), according to the definition of these expressions in all indeterminates \(x\).

Furthermore, by Theorems VII and II, after adjoining \(x_{i+1}\ldots x_n\), the highest elementary divisor \(E^{(i)}(x)\) is defined as the quotient \(\mathfrak M_{i-1}/\mathfrak G_{i-1}\). Thus, returning to polynomials in all indeterminates \(x\), for every \[\begin{equation} G(x)\equiv0(\mathfrak g_{i-1}) \quad\hbox{there is a}\quad b^{(i+1)}\ne0 \quad\hbox{such that}\quad b^{(i+1)}E^{(i)}(x)G(x)\equiv0(\mathfrak m). \tag{33} \end{equation}\] In particular \(\mathfrak g_0\) is the unit ideal, whence \[b^{(2)}E^{(1)}(x)\equiv0(\mathfrak m),\quad b^{(2)}\ne0, \qquad\hbox{and thus}\qquad E^{(1)}(x)\equiv0(\mathfrak g_1).\] In general, from (33), applied to \[E^{(1)}(x)\cdots E^{(i-1)}(x)\equiv0(\mathfrak g_{i-1}),\] there exists \(b^{(i+1)}\ne0\) such that \(b^{(i+1)}E^{(1)}\cdots E^{(i)}\equiv0(\mathfrak m)\); hence \(E^{(1)}\cdots E^{(i)}\equiv0(\mathfrak g_i)\). Since \(\mathfrak g_n=\mathfrak m\), one obtains \[\begin{equation} E_\mathfrak m=E^{(1)}\cdots E^{(n)}\equiv0(\mathfrak m) \quad\hbox{and consequently}\quad R_\mathfrak m=R^{(1)}\cdots R^{(n)}\equiv0(\mathfrak m). \tag{34} \end{equation}\]

If in (33) \(G(x)\) runs through the finitely many polynomials of an ideal basis of \(\mathfrak g_{i-1}\), and if \(c^{(i+1)}(x)\) denotes the product of the corresponding \(b^{(i+1)}(x)\), then \[c^{(i+1)}E^{(i)}(x)\mathfrak g_{i-1}\equiv0(\mathfrak m); \qquad c^{(i+1)}\ne0, \quad\hbox{and hence}\quad E^{(i)}(x)\mathfrak g_{i-1}\equiv0(\mathfrak g_i),\] and from this, as above, by finite repetition, \[E^{(n)}(x)\cdots E^{(i)}(x)\mathfrak g_{i-1}\equiv0(\mathfrak m)\] and consequently also \[R^{(n)}(x)\cdots R^{(i)}(x)\mathfrak g_{i-1}\equiv0(\mathfrak m).\] This proves all parts of Theorem VIII.

Theorem VIII depends essentially on properties of the elementary-divisor form; the characteristic significance of the resultant form for \(\mathfrak m\) is shown by

Theorem IX. If \(\mathfrak n\) is a divisor of \(\mathfrak m\) – divisor understood in the ideal sense – and if the resultant forms \(R_\mathfrak n\) and \(R_\mathfrak m\) agree, then the ideals \(\mathfrak n\) and \(\mathfrak m\) agree.

Let \(\mathfrak g_0\ldots\mathfrak g_{n-1},\mathfrak g_n=\mathfrak m\) and \(\mathfrak h_0\ldots\mathfrak h_{n-1},\mathfrak h_n=\mathfrak n\) denote the fundamental ideals of \(\mathfrak m\) and \(\mathfrak n\). First \(\mathfrak g_0\) and \(\mathfrak h_0\) are both the unit ideal, hence \(\mathfrak g_0=\mathfrak h_0\). Assume now that \(\mathfrak g_{i-1}=\mathfrak h_{i-1}\), so that \(\mathfrak M_{i-1}\) and \(\mathfrak N_{i-1}\) have the same fundamental module \(\mathfrak G_{i-1}\). The hypothesis \(R_\mathfrak n=R_\mathfrak m\), after adjoining \(x_{i+1}\ldots x_n\), may be written as \[N(\mathfrak G_{i-1}\mid\mathfrak N_{i-1})=N(\mathfrak G_{i-1}\mid\mathfrak M_{i-1}) \quad\hbox{or}\quad N(\mathfrak G^*_{i-1}\mid\mathfrak N^*_{i-1})=N(\mathfrak G^*_{i-1}\mid\mathfrak M^*_{i-1}).\] Here, in accordance with (32), \(\mathfrak N_{i-1}=(\mathfrak N^*_{i-1},\mathfrak G_{i-1})\), so that \(\mathfrak N^*_{i-1}\) is likewise a module of linear forms in \(\xi\), with \(\mathfrak G^*_{i-1}\) as its fundamental module. Since \(\mathfrak M_{i-1}\) is divisible by \(\mathfrak N_{i-1}\), which entails the divisibility of \(\mathfrak M^*_{i-1}\) by \(\mathfrak N^*_{i-1}\), Theorem IV shows that, under this adjunction, the two modules \(\mathfrak M^*_{i-1}\) and \(\mathfrak N^*_{i-1}\) agree, and consequently so do \(\mathfrak N_{i-1}\) and \(\mathfrak M_{i-1}\). But adjoining \(x_{i+1}\ldots x_n\) means that one adds to \(\mathfrak M_{i-1}\), respectively to \(\mathfrak m\), all polynomials \(G(x)\) for which \[b^{(i+1)}(x)G(x)\equiv0(\mathfrak m),\qquad b^{(i+1)}\ne0.\] Thus, by the adjunction, \(\mathfrak M_{i-1}\), respectively \(\mathfrak m\), passes into \(\mathfrak g_i\), and correspondingly \(\mathfrak n\) into \(\mathfrak h_i\); therefore \(\mathfrak g_i=\mathfrak h_i\). Hence finally \(\mathfrak g_n=\mathfrak h_n\), i.e. \(\mathfrak m=\mathfrak n\).

Finally the same transfer principle, applied to Theorem V, gives the following.

Theorem X. Let \(R^{(i)}=S^{(i)}T^{(i)}\) be a decomposition of \(R^{(i)}\) into relatively prime polynomials \(S^{(i)}\) and \(T^{(i)}\) in the polynomial sense. Then there exists one and only one ideal \(\mathfrak j\), and likewise one and only one ideal \(\mathfrak t\), both divisors of \(\mathfrak m\) with the same fundamental ideal of the \((i-1)\)-st stage, \(\mathfrak g_{i-1}\), such that \(S^{(i)}=N(\mathfrak G_{i-1}\mid\mathfrak S_{i-1})\) and \(T^{(i)}=N(\mathfrak G_{i-1}\mid\mathfrak T_{i-1})\). The ideals \(\mathfrak j\) and \(\mathfrak t\) are defined as the totality of polynomials \(S(x)\) and \(T(x)\), respectively, such that \[s^{(i+1)}(x)T^{(i)}(x)S(x)\equiv0(\mathfrak m),\quad s^{(i+1)}\ne0,\] \[t^{(i+1)}(x)S^{(i)}(x)T(x)\equiv0(\mathfrak m),\quad t^{(i+1)}\ne0,\] and \(\mathfrak g_i\) becomes the least common multiple of \(\mathfrak j\) and \(\mathfrak t\), \[\mathfrak g_i=[\mathfrak j,\mathfrak t].\]

It remains only to note that \(s^{(i+1)}\) and \(t^{(i+1)}\) can be chosen fixed for \(\mathfrak j\) and \(\mathfrak t\), as the product of the \(s^{(i+1)}\) and \(t^{(i+1)}\) corresponding to the basis elements of \(\mathfrak j\) and \(\mathfrak t\), and that, as remarked above, by adjoining \(x_{i+1}\ldots x_n\) the ideal \(\mathfrak m\) is transformed into \(\mathfrak g_i\). In particular, the decomposition of \(R^{(i)}\) can be continued up to powers of irreducible polynomials – primary factors – which entails a corresponding representation of \(\mathfrak g_i\) as a least common multiple.

§ 6. Proper elimination theory.

By Theorem VIII, respectively formula (34), the resultant form and the elementary-divisor form are divisible by \(\mathfrak m\), and therefore vanish at all zeros of \(\mathfrak m\). Here by “zeros of \(\mathfrak m\)” are meant value systems of \(x_1\ldots x_i\) belonging to the algebraically closed field derived from \(P(u;x_{i+1}\ldots x_n)\)¹⁶, such that every polynomial from \(\mathfrak m\) vanishes at these value systems, while \(x_{i+1}\ldots x_n\) are thought of as adjoined to the coefficient field \((i=1, \ldots,n)\). These adjoined \(x_{i+1}\ldots x_n\) can, as will be shown, also be replaced by quantities from \(P(u)\). The content of elimination theory, conversely, is the derivation of the zeros of \(\mathfrak m\) from those of the resultant form. This converse rests on the successive elimination to be developed in this section; since in the next section the divisibility of the form \(D^{(i)}(x)\) occurring in it by \(E^{(i)}(x)\) will be proved, the desired converse follows from the divisibility of a power of \(E^{(i)}(x)\) by \(R^{(i)}(x)\).

The theory of successive elimination to be given here rests on the following result.

Theorem XI. Let \(\mathfrak b\) be an ideal of polynomials in one indeterminate \(t\) with coefficients in \(P\), hence a principal ideal; let \(h(t)\) be a polynomial from \(\mathfrak b\) of degree \(k\), and let \(\mathfrak B\) be the module of linear forms in \(1,t,\ldots,t^{k-1}\) into which \(\mathfrak b\) passes modulo \(h(t)\). Then \(\mathfrak B\) has rank \(k-p\), where \(p\) is the degree of the basis polynomial \(f(t)\) of \(\mathfrak b\).

By hypothesis \(h(t)=h_1(t)f(t)\), where \(h_1(t)\) has degree \(k-p\). The residue classes of \(\mathfrak b\) modulo \(h(t)\) represented by \(f(t),tf(t),\ldots,t^{k-p-1}f(t)\) are therefore linearly independent; and every residue class of \(\mathfrak b\) modulo \(h(t)\) can be represented linearly by these \(k-p\) special classes, with coefficients in \(P\). But modulo \(h(t)\) the ideal \(\mathfrak b\) passes into a module of linear forms in \(1,t, \ldots,t^{k-1}\), whose rank is therefore exactly \(k-p\).

Now let \(\mathfrak a=\mathfrak a_1\) be a transformed ideal of polynomials, and let \(A^{(1)}(x)\) be a polynomial from \(\mathfrak a_1\), regular in \(x_1\), of degree \(k_1\). Let \(\mathfrak A_1\) be the module of linear forms in \(1,x_1, \ldots,x_1^{k_1-1}\), with polynomials \(a^{(2)}(x)\) as coefficients, into which \(\mathfrak a_1\) passes modulo \(A^{(1)}(x)\). Let \(\mathfrak a_2\) denote the ideal in \(x_2\ldots x_n\) derived from all \(k_1\)-rowed determinants from \(\mathfrak A_1\). By Theorem XI, \(\mathfrak a_2\) is the zero ideal if and only if the polynomials from \(\mathfrak a_1\) have a common divisor, in the polynomial sense, of degree \(p_1>0\) in \(x_1\). If \(\mathfrak a_2\) is different from the zero ideal, then, since \(\mathfrak a\) is assumed transformed, it contains a polynomial \(A^{(2)}(x)\) regular in \(x_2\), of degree \(k_2\), as is seen directly by composing the transformation (12) from the special transformations (25) and (26). Let \(\mathfrak A_2\) again denote the module of linear forms in \(1,x_2, \ldots,x_2^{k_2-1}\) into which \(\mathfrak a_2\) passes modulo \(A^{(2)}(x)\), and \(\mathfrak a_3\) the ideal in \(x_3\ldots x_n\) derived from all \(k_2\)-rowed determinants from \(\mathfrak A_2\); this ideal must contain a polynomial \(A^{(3)}(x)\) regular in \(x_3\).

In this way one continues the process until one reaches a zero ideal, or until \(\mathfrak a_{n+1}\) becomes the unit ideal; for since \(\mathfrak a_{n+1}\) is free of the \(x\), it can only be the zero ideal or the unit ideal. It is seen that the ideals \(\mathfrak a_1,\mathfrak a_2, \ldots\) are uniquely determined by \(\mathfrak a\), independently of the chosen polynomials \(A^{(\lambda)}(x)\). This is clear for \(\mathfrak a_1=\mathfrak a\), and can therefore be assumed for \(\mathfrak a_1, \ldots,\mathfrak a_i\). If \(\mathfrak a_i\) passes modulo \(A^{(i)}(x)\) into the module of linear forms \(\mathfrak A_i\), then \(\mathfrak A_i\) can also be characterized as the totality of the polynomials from \(\mathfrak a_i\) that do not reach degree \(k_i\) in \(x_i\); thus \(\mathfrak A_i\) depends only on the degree \(k_i\) of \(A^{(i)}(x)\). Let \(\bar A^{(i)}(x)\) be a polynomial from \(\mathfrak a_i\), regular in \(x_i\), of degree \(\bar k_i>k_i\), let \(\overline{\mathfrak A}_i\) be the corresponding module of linear forms, and let \(\bar{\mathfrak{a}}_{i+1}\) be the ideal derived from the \(\bar k_i\)-rowed determinants from \(\overline{\mathfrak A}_i\). Then the module equation holds: \[\overline{\mathfrak A}_i=(\mathfrak A_i,A^{(i)},x_iA^{(i)},\ldots,x_i^{\bar k_i-k_i-1}A^{(i)}).\] Since, by the regularity of \(A^{(i)}(x)\) in \(x_i\), the polynomials \(A^{(i)},x_iA^{(i)}, \ldots\) can be introduced as new indeterminates of the linear forms, it follows that the \(k_i\)-rowed determinants from \(\mathfrak A_i\) agree with the \(\bar k_i\)-rowed determinants from \(\overline{\mathfrak A}_i\), and therefore \(\bar{\mathfrak{a}}_{i+1}=\mathfrak a_{i+1}\).¹⁷

Furthermore \(\mathfrak a_{i+1}\) is always divisible by \(\mathfrak a_i\). This is clear if \(\mathfrak a_{i+1}\) is the zero ideal; in the opposite case \(\mathfrak A_i\) has rank \(k_i\), so that \(\mathfrak a_{i+1}\) is the ideal which by definition consists of the \(k_i\)-rowed determinants from \(\mathfrak A_i\), and is divisible by \(\mathfrak A_i\) and hence by \(\mathfrak a_i\). Thus every \(\mathfrak a_{i+1}\) is divisible by \(\mathfrak a\); if therefore \(\mathfrak a_{n+1}\) becomes the unit ideal, then \(\mathfrak a\) itself is the unit ideal.

Let now \(\mathfrak a\) be different from the unit ideal, with \(\mathfrak a_1\ne0, \ldots,\mathfrak a_i\ne0\), and \(\mathfrak a_{i+1}=0\). Let \(D^{(i)}(x)\) be the greatest common divisor, existing by Theorem XI, of all polynomials from \(\mathfrak a_i\), where divisor is understood in the polynomial sense; as a divisor of \(A^{(i)}(x)\), \(D^{(i)}(x)\) is itself regular in \(x_i\) of degree \(p_i>0\). Adjoin \(x_{i+1}\ldots x_n\) to \(P(u)\); then \(D^{(i)}(x)\) has, in an algebraic extension field of \(P(u;x_{i+1}\ldots x_n)\), a decomposition into \(p_i\) linear factors. Let \(x_i-\bar x_i\) be one such linear factor, so that all polynomials from \(\mathfrak a_i\) vanish for \(x_i=\bar x_i\); then, by Theorem XI, the polynomials from \(\mathfrak a_{i-1}\) acquire for this specialization a greatest common divisor \(D^{(i-1)}(x)\). As a divisor of \(A^{(i-1)}(x)\), \(D^{(i-1)}(x)\) is again regular in \(x_{i-1}\) of degree \(p_{i-1}>0\); in particular it cannot vanish identically, and in an extension field of \(P(u, \bar x_i,x_{i+1}, \ldots,x_n)\) it decomposes into \(p_{i-1}\) linear factors. If \(x_{i-1}-\bar x_{i-1}\) is such a linear factor, there corresponds to it a greatest common divisor from \(\mathfrak a_{i-2}\). Continuing in this way, one reaches finitely many common zeros of \(\mathfrak a\) lying in an algebraic extension field of \(P(u;x_{i+1}\ldots x_n)\). Conversely, because \(\mathfrak a_i\) is divisible by \(\mathfrak a\), every zero of \(\mathfrak a\) is also a zero of \(\mathfrak a_i\). If, in particular, \(x_1=\bar x_1, \ldots,x_i=\bar x_i,x_{i+1}=x_{i+1}, \ldots,x_n=x_n\) is a zero of \(\mathfrak a\), then \(\mathfrak a_{i+1}=0\) follows, and the polynomials from \(\mathfrak a_i\) acquire a greatest common divisor with the linear factor \(x_i-\bar x_i\). Summarizing:

Theorem XII. Let \(\mathfrak a\) be different from the unit ideal, with \(\mathfrak a_1\ne0, \ldots,\mathfrak a_i\ne0\) and \(\mathfrak a_{i+1}=0\). Then, after adjoining \(x_{i+1}\ldots x_n\) to \(P(u)\), there exist finitely many associated value systems \(\bar x_1\ldots\bar x_i\), lying in an algebraic extension field of \(P(u,x_{i+1}, \ldots,x_n)\), such that all polynomials from \(\mathfrak a\) vanish for \(x_1=\bar x_1, \ldots,x_i=\bar x_i,x_{i+1}=x_{i+1}, \ldots,x_n=x_n\). Conversely, if such a zero of \(\mathfrak a\) is present, then \(\mathfrak a_{i+1}=0\) follows, as does the occurrence of a common divisor \(D^{(i)}(x)\) of all polynomials from \(\mathfrak a_i\) that has the linear factor \(x_i-\bar x_i\).

Theorem XII shows in particular that every ideal \(\mathfrak a\) having no zero is the unit ideal.

To carry out a decomposition that also explicitly exhibits the association of the value systems, introduce new indeterminates \(v\), which may be adjoined to \(P(u,x_{i+1}, \ldots,x_n)\). Put \[\begin{equation} z_i=-v_1x_1-\cdots-v_{i-1}x_{i-1}+x_i. \tag{35} \end{equation}\] Then the composition of (12) with (35) is again a substitution of type (12), in which now only the \(u_{ik}\) are replaced by \(w_{ik}=u_{ik}-v_k\), and more generally \(u_{i+h,k}\) by \(w_{i+h,k}=u_{i+h,k}-u_{i+h,i}v_k\). If the substitution (35) carries the, by hypothesis transformed, ideal \(\mathfrak a\) into \(\mathfrak b\), then \(\mathfrak b\) is obtained from \(\mathfrak a\) simply by replacing the \(u_{\mu\nu}\) by \(w_{\mu\nu}\) and adjoining the \(v\); and by the same process the ideals \(\mathfrak b_1,\mathfrak b_2, \ldots\) defined by \(\mathfrak b\) arise from \(\mathfrak a_1, \mathfrak a_2, \ldots\). Conversely \(\mathfrak a_i\) is obtained from \(\mathfrak b_i\) by \(v=0\); therefore the ideals \(\mathfrak a_i\) and \(\mathfrak b_i\) are simultaneously zero ideals or simultaneously different from the zero ideal.

Now again let \(\mathfrak a_1\ne0; \ldots;\mathfrak a_i\ne0;\mathfrak a_{i+1}=0\), and hence also \(\mathfrak b_1\ne0; \ldots;\mathfrak b_i\ne0;\mathfrak b_{i+1}=0\). Let \(\bar x_1\ldots\bar x_i,x_{i+1}\ldots x_n\) be an associated zero-system of \(\mathfrak a\). Define \(\bar z_i=\bar x_i+v_1\bar x_1+\cdots+v_{i-1}\bar x_{i-1}\). Then \(\bar x_1\ldots\bar x_{i-1},\bar z_i,x_{i+1}\ldots x_n\) is, by (35), a zero of \(\mathfrak b\). Hence if \(H^{(i)}(z,x)\) denotes the greatest common divisor of all polynomials from \(\mathfrak b_i\), which may be assumed integral and primitive in the \(v\), then by the last part of Theorem XII, \(H^{(i)}(z,x)\) has the factor \[z_i-\bar z_i=z_i-(\bar x_i+v_1\bar x_1+\cdots+v_{i-1}\bar x_{i-1}),\] and to every zero of \(\mathfrak a\) occurring after adjoining \(x_{i+1}\ldots x_n\) there corresponds such a factor. It must be shown that this exhausts the factorization of \(H^{(i)}(z,x)\), i.e. that no factor \(H^{(i)}_1\) can be separated from \(H^{(i)}\) which does not decompose into linear factors in \(z\) and \(v\). If this were the case, then because of the assumed primitivity in \(v\), \(H^{(i)}\) would contain at least one linear factor \(z_i-\bar z_i\), where \(\bar z_i\) belongs to an algebraic extension field of \(P(u,v,x_{i+1}, \ldots,x_n)\). If \(\bar x_1\ldots\bar x_{i-1},\bar z_i,x_{i+1}\ldots x_n\) is the corresponding zero of \(\mathfrak b\), then it must coincide with one of the finitely many zeros of \(\mathfrak a\) that occur after adjoining \(x_{i+1}\ldots x_n\), and so it must be independent of \(v\). Hence \(\bar z_i\) necessarily contains the \(v\) linearly, not algebraically or rationally non-linearly; contrary to the assumption, another linear factor \(z_i-(\bar x_i+v_1\bar x_1+\cdots+v_{i-1}\bar x_{i-1})\) in \(z\) and \(v\) can be split from \(H^{(i)}_1\). Thus the explicit decomposition is \[H^{(i)}(z,x)=\prod \{z_i-(\bar x_i+v_1\bar x_1+\cdots+v_{i-1}\bar x_{i-1})\}^{\alpha_\lambda}.\] Since the degree of \(H^{(i)}\) and the individual exponents \(\alpha_\lambda\) must agree with the corresponding ones for \(D^{(i)}(x)\) – for \(H^{(i)}\) arises from \(D^{(i)}\) by the specialization \(u_{\mu\nu}=w_{\mu\nu}\), and \(D^{(i)}\) from \(H^{(i)}\) by the specialization \(v=0\) – this decomposition also shows that, because the indeterminates \(u_{\mu\nu}\) are adjoined to \(P\), every zero \(\bar x_i\) of \(D^{(i)}\) occurring in the elimination starting from \(\mathfrak a\) corresponds to a greatest common divisor from \(\mathfrak a_{i-1}\ldots\mathfrak a_1\) that is respectively linear, or to a power of such; hence in each case only one associated zero-system \(\bar x_1\ldots\bar x_i,x_{i+1}\ldots x_n\) of \(\mathfrak a\) is obtained.¹⁸

§ 7. Resultant form and elimination theory.

To establish the connection between the resultant form and elimination theory, apply the successive elimination of § 6 specifically to the quotient \(\mathfrak a=\mathfrak m/\mathfrak g_{i-1}\) of ideal by fundamental ideal of the \((i-1)\)-st stage. It must first be shown that this quotient is a transformed ideal. Now \(\mathfrak m\) is transformed and hence, by Theorem VI of § 3, so is \(\mathfrak g_{i-1}\). From \[H(x)\equiv0(\mathfrak m/\mathfrak g_{i-1});\qquad H(x)=\overline H(y)=\sum\alpha_i(u)\overline H_i(y)=\sum\alpha_i(u)H_i(x)\] one obtains \[\overline H(y)\overline{\mathfrak{g}}_{i-1,(u)}\equiv0(\overline{\mathfrak{m}}_{(u)}), \quad\hbox{hence also}\quad \overline H(y)\overline{\mathfrak{g}}_{i-1}\equiv0(\overline{\mathfrak{m}}_{(u)}),\] or \[\overline H_i(y)\overline{\mathfrak{g}}_{i-1}\equiv0(\overline{\mathfrak{m}}), \quad\hbox{hence}\quad H_i(x)\mathfrak g_{i-1}\equiv0(\mathfrak m).\] This proves the divisibility of \(H_i(x)\) by \(\mathfrak m/\mathfrak g_{i-1}\) and hence that this ideal is transformed, so that the elimination method of § 6 applies.

But (30), taking account of (28) – § 4 – shows that \(C^{(i)}(x)\) is divisible by \(\mathfrak m/\mathfrak g_{i-1}\), where \(C^{(i)}(x)\) denotes any non-identically vanishing determinant of degree \(\rho\) from \(\mathfrak M^*_{i-1}\). Since for \(i>1\) this \(C^{(i)}\) has degree zero in \(x_1\), the module \(\mathfrak A_1\) corresponding to the ideal \(\mathfrak a=\mathfrak a_1\) contains the polynomials \(C^{(i)},x_1C^{(i)}, \ldots,x_1^{k_1-1}C^{(i)}\), and therefore \((C^{(i)})^{k_1}\) is divisible by \(\mathfrak a_2\); similarly \((C^{(i)})^{k_1k_2}\) is divisible by \(\mathfrak a_3\), and finally \((C^{(i)})^{k_1k_2\cdots k_{i-1}}\) by \(\mathfrak a_i\). Thus \(\mathfrak a_1, \ldots,\mathfrak a_i\) are different from the zero ideal, as also holds for \(i=1\). Let \(D^{(i)}(x)\) now be the greatest common divisor of all polynomials from \(\mathfrak a_i\), which has degree \(p_i>0\) only when \(\mathfrak a_{i+1}\) is the zero ideal. By the definition of \(D^{(i)}(x)\), and taking into account the divisibility of \(\mathfrak a_i\) by \(\mathfrak a\), one obtains \[d^{(i+1)}D^{(i)}(x)\equiv0(\mathfrak m/\mathfrak g_{i-1});\qquad d^{(i+1)}\ne0.\] Thus \(d^{(i+1)}D^{(i)}\) is a polynomial, free of \(x_1\ldots x_{i-1}\), belonging to \(\mathfrak m/\mathfrak g_{i-1}\). But \(E^{(i)}(x)\) was defined (Theorem II and § 4), as the highest elementary divisor of \(\mathfrak M_{i-1}\) or \(\mathfrak M^*_{i-1}\), to be the greatest common divisor, in the polynomial sense, of all polynomials from \(\mathfrak m/\mathfrak g_{i-1}\) that are free of \(x_1\ldots x_{i-1}\). Therefore \(d^{(i+1)}D^{(i)}\), and because of the regularity of \(E^{(i)}(x)\) in \(x_i\) also \(D^{(i)}(x)\), is divisible by \(E^{(i)}(x)\).

The same argument can be carried out if the transformation (12) had from the beginning been composed with (35), that is, if the indeterminates \(u_{\mu\nu}\) had been replaced by \(w_{\mu\nu}\); this gives the divisibility of \(H^{(i)}(z,x)\) by the corresponding \(\bar E^{(i)}(z,x)\). Since, just as \(D^{(i)}(x)\) and \(H^{(i)}(z,x)\) pass into one another by specialization, so too do \(E^{(i)}(x)\) and \(\bar E^{(i)}(z,x)\), and also \(R^{(i)}(x)\) and \(\bar R^{(i)}(z,x)\), the multiplicity numbers in the decomposition into linear factors must also agree mutually. Taking into account, furthermore, that a power of \(E^{(i)}(x)\) is divisible by \(R^{(i)}(x)\), one obtains the following summary.

Theorem XIII. If \(R^{(i)}(x)\) is decomposed in an algebraic extension field of \(P(u,x_{i+1}, \ldots,x_n)\) into linear factors \(x_i-\bar x_i\), then \(\bar x_i\) can be completed in one and only one way to an associated value system \(\bar x_1\ldots\bar x_i,x_{i+1}\ldots x_n\) that is a zero of \(\mathfrak m/\mathfrak g_{i-1}\) and consequently of \(\mathfrak m\). The finitely many zeros of \(\mathfrak m\) arising in this way from the linear factors of \(R^{(i)}\) – finitely many after adjoining \(x_{i+1}\ldots x_n\) – are collected by the explicit decomposition \[\begin{equation} \bar R^{(i)}(z,x)=\prod\{z_i-(\bar x_i+v_1\bar x_1+\cdots+v_{i-1}\bar x_{i-1})\}^{\alpha_\lambda}. \tag{36} \end{equation}\] Because of the regularity of \(\bar R^{(i)}(z,x)\) in \(z_i\), this decomposition is preserved if the \(x_{i+1}\ldots x_n\) adjoined to \(P(u)\) are replaced by arbitrary special value systems \(\bar x_{i+1}\ldots\bar x_n\) from \(P(u)\) or from the algebraically closed field derived from it.

This also accomplishes a separation of the zeros of \(\mathfrak m\) according to the individual factors of the resultant form; the number of indeterminates \(x\) adjoined to \(P(u)\) is also called the dimension of the corresponding algebraic object.

If, in the decomposition of \(\bar R^{(i)}(z,x)\) or of the corresponding \(R^{(i)}(x)\), the factors conjugate over \(P(u,x_{i+1}\ldots x_n)\) are collected together – factors that for general \(P\) may be wholly or partially identical – one obtains a decomposition of \(R^{(i)}(x)\) into powers of polynomials irreducible over \(P(u,x_{i+1}\ldots x_n)\): \[R^{(i)}(x)=S^{(i)}_1(x)^{\beta_1}\cdots S^{(i)}_\nu(x)^{\beta_\nu}.\] By Theorem X, this decomposition corresponds to a representation \(\mathfrak g_i=[\mathfrak j_1\ldots\mathfrak j_\nu]\) such that \(S^{(i)}_\mu(x)^{\beta_\mu}\) is equal to the norm of \(\mathfrak g_{i-1}\) with respect to the ideal \(\mathfrak j_\mu\), respectively to the norm of the module \(\mathfrak G_{i-1}\) with respect to the module corresponding to \(\mathfrak j_\mu\). This clarifies the meaning of the exponents \(\beta_\mu\): in particular, if \(\gamma_\mu\) is the degree of \(S^{(i)}_\mu\), then the multiplicity \(\beta_\mu\gamma_\mu\) is equal to the number of residue classes of \(\mathfrak g_{i-1}\) modulo \(\mathfrak j_\mu\) that are linearly independent over \(P(u,x_{i+1}\ldots x_n)\).

Finally consider briefly the special case where \(P\) has characteristic zero. If the \(u_{\mu\nu}\) are specialized to quantities \(\bar u_{\mu\nu}\) from \(P\) in such a way that the \(n\) determinants \(C^{(i)}(x)\) \((i=1,2, \ldots,n)\) remain regular in \(x_i\),¹⁹ then the regularity of \(R^{(i)}\) is also preserved. Under this specialization, \(R^{(i)}\) and likewise \(\bar R^{(i)}(z,x)\) become a common divisor of all \(\rho\)-rowed determinants from \(\mathfrak M^*_{i-1}\), though not necessarily the greatest common divisor. But the specialization can always be chosen so that this latter property also remains true. For, as the existence of the module basis shows, \(R^{(i)}(z,x)\) can also be characterized as the greatest common divisor of finitely many \(\rho\)-rowed determinants from \(\mathfrak M^*_{i-1}\). The decomposition of the so specialized \(\bar R^{(i)}(z,x)\) is then obtained simply by replacing the \(u_{\mu\nu}\) by \(\bar u_{\mu\nu}\) in (36), so that the whole elimination theory is preserved.