Galois Theory

Proposition 1.3. Let φ be an automorphism of a field extension K/F , and f(x) ∈ F [x]. Let α1, . . . , αn be the roots of f(x) lying in K. Then φ permutes the set {α1, . . . , αn}. If also the set of αi generate K over F , then two automorphisms φ1, φ2 of K/F which agree on all the αi are equal. Thus, in this case we have an inclusion of Aut(K/F ) as a subgroup of Sym({α1, . . . , αn}) ∼= Sn. Proof. For the first part, it suffices to observe that if α ∈ K is a root of f(x), then because φ fixes F and is a homomorphism, φ(α) is also a root of f(x). For the second part, because every element of K can be written as a rational function in the αis, with coefficients in F , if both φ1 and φ2 keep F fixed and have the same values on all the αi, we conclude that they agree on all of K.