Local fields

Twelve lessons on valuations, completions, Hensel lifting, local extensions, ramification, unit groups and explicit examples, with proofs and complete exercise solutions.

Prerequisites: graduate algebra, real analysis and the measure and topology results named precisely in each lesson. Earlier programme proof providers are identified by exact title and result; this release contains the NT-LOC lessons.

1. Absolute values, valuations and Ostrowski's theorem
2. Completions, the p-adic numbers and complete discretely valued fields
3. Hensel's lemma, squares and roots of unity in p-adic fields
4. Extensions of complete valued fields
5. Places of number fields in extensions and the product formula
6. Local fields: classification and Haar measure
7. Unramified and totally ramified extensions
8. Tame ramification and the tame Galois group
9. Ramification groups and the different of a local extension
10. Herbrand's function and the upper numbering
11. The multiplicative group of a local field
12. Cyclotomic, quadratic and Kummer extensions of local fields

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Original text: CC0.