Circulant Hadamard matrices

OpenAI's proof (2026) of the circulant Hadamard conjecture, usually attributed to Ryser: real circulant Hadamard matrices exist only in orders 1 and 4. The course works in the group ring of the cyclic group: it proves the local structure of cyclotomic integers near a prime and Turyn's restriction to orders 4u² with u odd, shows that alternating products of character values are roots of unity of odd order, and derives a contradiction modulo a prime above 2; even-length Barker sequences follow.

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Circulant Hadamard matrices and group rings
2. Local rings of cyclotomic integers
3. Alternating character products
4. The circulant Hadamard conjecture

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Rings, ideals, quotients, localization and the Chinese remainder theorem (core course Abstract Algebra II); diagonalization of commuting matrices of finite order (core course Linear Algebra).