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Algebraic topology: a primer by Deo S.

By Deo S.

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Concerning the decomposition p = PZ A in KZ we should say that, apart from PZ , there need not exist g − 1 other different factors: the prime ideals 38 1. The Theory of Galois Extensions PσZ belonging to the conjugates Pσ live in the conjugate fields KZσ , which need not at all coincide. The Hilbert field sequence for P only explains the facts concerning P and its origin from p, but not the complete decomposition of p in K. The situation is different for abelian extensions; then the conjugate fields coincide, and in the common decomposition fields KZ the prime ideal p has the form PσZ , p= (σ system of representatives for G/GZ ), σ because we can argue for each conjugate prime ideal as we have done for P.

We will now investigate how the Hilbert subgroup and subfield series behave under extension of the base field, that is, when we take a subextension E of K|F as our base field. Here we have, when we denote the Galois correspondence between fields and groups by ←→ : Theorem 27. If E ←→ H, then we get the subgroup series for K|E associated to P by taking the intersection with H of the original subgroup series; the corresponding subextensions arise by taking the compositum with E. Proof. We are given K ⊆ E ⊆ F.

Here we have, when we denote the Galois correspondence between fields and groups by ←→ : Theorem 27. If E ←→ H, then we get the subgroup series for K|E associated to P by taking the intersection with H of the original subgroup series; the corresponding subextensions arise by taking the compositum with E. Proof. We are given K ⊆ E ⊆ F. Let H denote the subgroup associated to E. g. the decomposition group GZ of P with respect to K|E. Since GZ fixes P as well as the elements of E, we see that GZ is the biggest subgroup contained both in GZ and H, and this is the intersection GZ ∩ H.

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