Today, we are diving into a classic exercise from Joseph A. Gallian's "Contemporary Abstract Algebra" (5th Edition, Chapter 32, Problem 1). We are going to look at the formal proof first, and then we will unpack exactly what it means for those who might be new to abstract algebra. Part 1: The Formal Proof The Prompt: Let E be an extension field of Q. Show that any automorphism of E acts as the identity on Q. The Proof: Let E be an extension field of the rational numbers Q, and let phi be an automorphism of E. By the definition of a field automorphism, phi is a bijection that preserves both addition and multiplication, and maps the multiplicative identity of E to itself. Therefore, phi(1) = 1. To show that phi acts as the identity on Q, we must first demonstrate that it fixes all integers. For any positive integer n, we can express n as the sum of n ones. Using the additive property of the homomorphism, we have: phi(n) = phi(1 + 1 + ... + 1) [n times] = phi(1) + phi(1) + ...