A Brief Introduction to Galois Theory: Galois Group of ℚ(√2) Over ℚ, Gal(ℚ(√2)/ℚ)
In Field Theory, let E be an extension field of F. An automorphism of E is a ring isomorphism from E to itself. The Galois group Gal(E/F) is the group (under function composition) of all automorphisms of E that fix F (send every element of F to itself). It turns out that Gal(ℚ(√2)/ℚ) = {ε, α}, where ε is the identity mapping ε(a+b√2)=a+b√2 and α is the "conjugate" mapping α(a+b√2)=a-b√2. This Galois group is isomorphic to cyclic group ℤ2. 🔴 "Contemporary Abstract Algebra", by Joe Gallian: https://amzn.to/2ZqLc1J
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