A galois correspondence for radical extensions of fields

Galois Groups of Radical Extensions

Theorem 1.1 (Kummer theory). Let m ∈ Z>0, and suppose that the subgroup μm(K) = {ζ ∈ K∗ : ζ = 1} of K∗ has order m. Write K∗1/m for the subgroup {x ∈ K̄∗ : x ∈ K∗} of K̄∗. Then K(K∗1/m) is the maximal abelian extension of exponent dividing m of K inside K̄, and there is an isomorphism Gal(K(K∗1/m)/K) ∼ −→ Hom(K∗, μm(K)) that sends σ to the map sending α to σ(β)/β, where β ∈ K∗1/m satisfies β = α.

Galois Extensions of Hilbertian Fields

We prove the following result: Theorem. Let K be a countable Hilbertian field, S a finite set of local primes of K, and e ≥ 0 an integer. Then, for almost all ∈ G(K)e, the field Ks[ ] ∩Ktot,S is PSC. Here a local prime is an equivalent class p of absolute values of K whose completion is a local field, K̂p. Then Kp = Ks ∩ K̂p and Ktot,S = T p∈S T σ∈G(K) K σ p . G(K) stands for the absolute Galois ...

Compositum of Galois Extensions of Hilbertian Fields

Procyclic Galois Extensions of Algebraic Number Fields

6 1 Iwasawa’s theory of Zp-extensions 9 1.

The Stable Galois Correspondence for Real Closed Fields

In previous work [7], the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if L/k is a nite Galois extension of elds with Galois group G, there is a functor c∗ L/k : SHG → SHk from the G-equivariant stable homotopy category to the stable motivic homotopy category over k such that c∗ L/k (G/H+) = Spec(L)+. The main theorem of [7] s...