Galois Cohomology [Lecture notes] by Seyfi Türkelli

By Seyfi Türkelli

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Let X be a set and Fp (X) be the free profinite p-group on X. Then, the homomorphisms of Fp (X) into a discrete p−group A in 1-1 correspondence with set theoretic maps X → A which send almost all x ∈ X to 0 in A. Consequently, cdp Fp (X) ≤ 1. Proof. Easy. 18. H 1 (Fp (X); Z/pZ) ∼ = (Z/pZ)|X| . Consequently, cdp (Fp (X)) = cd(Fp (X)) = 1. 19. Fp (X) ∼ = Fp (Y ) if and only if |X| = |Y |. 20. cdZp = cdp Zp = 1 and cd(Z) ←− 30 Chapter 4 Galois Cohomology: An Application Many deep results, especially about Brauer groups, can be established cohomological machinery.

17. Let X be a set and Fp (X) be the free profinite p-group on X. Then, the homomorphisms of Fp (X) into a discrete p−group A in 1-1 correspondence with set theoretic maps X → A which send almost all x ∈ X to 0 in A. Consequently, cdp Fp (X) ≤ 1. Proof. Easy. 18. H 1 (Fp (X); Z/pZ) ∼ = (Z/pZ)|X| . Consequently, cdp (Fp (X)) = cd(Fp (X)) = 1. 19. Fp (X) ∼ = Fp (Y ) if and only if |X| = |Y |. 20. cdZp = cdp Zp = 1 and cd(Z) ←− 30 Chapter 4 Galois Cohomology: An Application Many deep results, especially about Brauer groups, can be established cohomological machinery.

Then there exists α ∈ K such that K = k(α) and α satisfies the equation xp − x − a = 0 with some a ∈ k. ii. Conversely, given a ∈ k, the polynomial f (x) = xp − x − a either has one root in k, in which case all its roots are in k, or it is irreducible. In this latter case, is α is a root then k(α) is cyclic of degree p over k. Proof. Let σ be a generator of G. Clearly, the automatically-continuous map f : G → K + defined by f (σ r ) = r is a cocyle. , p − 1 or, in other words, σ r (α) = α + r. Therefore, α has p distinct conjugates and [k(α) : k] ≥ p; Thus K = k(α).

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