Preliminary Notes on Galois Theory
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چکیده
Definition 1.1. Let E be a field. An automorphism of E is a (ring) isomorphism from E to itself. The set of all automorphisms of E forms a group under function composition, which we denote by AutE. Let E be a finite extension of a field F . Define the Galois group Gal(E/F ) to be the subset of AutE consisting of all automorphisms σ : E → E such that σ(a) = a for all a ∈ F . We write this last condition as σ|F = Id. It is easy to check that Gal(E/F ) is a subgroup of AutE (i.e. that it is closed under composition, Id ∈ Gal(E/F ), and, if σ ∈ Gal(E/F ), then σ−1 ∈ Gal(E/F )). Note that, if F0 is the prime subfield of E (F0 ∼= Q or F0 ∼= Fp depending on whether the characteristic is 0 or a prime p), then AutE = Gal(E/F0). In other words, every σ ∈ AutE satisfies σ(1) = 1 and hence σ(a) = a for all a ∈ F0. If we have a sequence of fields F ≤ K1 ≤ K2 ≤ E, then Gal(E/K2) ≤ Gal(E/K1) ≤ Gal(E/F ) (the order is reversed). As with the symmetric group, we shall usually write the product in Gal(E/F ) as a product, i.e. the product of σ1 and σ2 is σ1σ2, instead of writing σ1 ◦σ2, and shall often write 1 for the identity automorphism Id. A useful fact, which is left as a homework problem, is that if E is a finite extension of a field F and σ : E → E is a ring homomorphism such that σ(a) = a for all a ∈ F , then σ is surjective, hence an automorphism, hence is an element of Gal(E/F ).
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