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How to determine the splitting type of a prime

By Melanie Matchett Wood


Let E/K be an extension of number fields (or function fields?) of degree n, and let p be a prime of K. The goal of these notes is to explain how to determine the splitting type of p in E via information about decomposition and inertia groups. Many texts only treat the case when E/K is Galois, except for the section on “Factorization in Nonnormal Extensions ” in [1, Chapter III, Section 2], which unfortunately only covers the unramified case. Let L be the Galois closure of E/K. Let G = Gal(L/K) ⊂ Sn vis its permutation action on the n homomorphisms E → L. Let H the subgroup of G corresponding to E via Galois theory (which is the intersection of G with the Sn−1 ⊂ Sn stabilizing the identity inclusion E → L). If you weren’t sure how G acts on the n homomorphisms E → L, then I’ve just told you, because a transitive action is given by any stabilizer, and thus the permutation action G ⊂ Sn above is just the action of G on cosets of H. We use H\G to denote the cosets Hτ of H, but it is important to remember that H is not normal. Let ℘ be a prime of L above p. Let G(℘) ⊂ G be the decomposition group of ℘ and I(℘) ⊂ G(℘) be the inertia group. The group G(℘) acts on H\G (this action is given by the permutation representation G(℘) ⊂ G ⊂ Sn), and the primes of E above p correspond t

Year: 2011
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