Notes on Algebra (fields)

Proof. The intersection P of all subfields of F is a field by Exercise 1.4. Consider the ring homomorphism φ : Z → F given by φ(n) = n · 1. Since any subfield contains 1 and is closed under addition, imφ is contained in P . If Char F = p 6= 0 then imφ is isomorphic to Z/pZ = Fp. Since this is a field, we have P = imφ ∼= Fp. If Char F = 0 then φ is injective. Define φ̂ : Q → F by φ̂(m/n) = φ(m)/φ(n) for any m, n ∈ Z with n 6= 0. It is easy to check that φ̂ is well-defined, and is an injective homomorphism. Moreover, φ̂(Q) ⊆ P since P is closed under the field operations. Thus P = im φ̂ ∼= Q if Char F = 0.