bibkey: milne2022fields
authors: J. S. Milne
year: 2022
title: "Fields and Galois Theory, version 5.10"
doi: null
url: https://www.jmilne.org/math/CourseNotes/FT.pdf
claim: "Theorem 5.30, Example 5.31(b), and Remark 5.32 give the classical Kummer correspondence and rational square classes underlying finite prime-radical fields."
strata_touched:
  - D5/S3/Factorization/Galois/QuadraticPrimeRadicalKernel
license: citation-only
triage: anchor

Classical multiquadratic square classes

Locator

Theorem 5.30, Example 5.31(b), and Remark 5.32, printed pages 75–76, describe the Kummer correspondence, its degree and pairing, and rational square classes represented by signed products of distinct primes. The specialization used here is exponent two over the rationals, which contain the primitive second root of unity minus one.

The formal module uses positive rational primes as radicands and the actual intermediate field of the complex numbers generated by their positive real square roots. Its scalar is an arbitrary rational number, including zero and negative values. It proves the rational square kernel and exact degree by simultaneous induction and an explicit quadratic-model range equivalence. It does not claim the full Kummer correspondence or a novelty result.

The source is Milne, Fields and Galois Theory, version 5.10, at https://www.jmilne.org/math/CourseNotes/FT.pdf?source=1. The PDF SHA-256 is 5c43ea0bf4ec190b819727fe720414a80e09f9e762533d08bad5b8a0b6de7273. This is a citation-only reference. Project and Mathlib source remain under their existing licenses.

The original GPC5 arithmetic Frobenius and inertia statements and the original GPC7 exact density comparison are outside this module. Sharifi’s Algebraic Number Theory, Proposition 7.2.1 and Corollary 7.2.3, remain the separate sources for the density statements; they are not represented as theorems proved here.