Lemma for the Fundamental Theorem of Galois Theory

Lemma for FTGT By Susam Pal on 09 Mar 2025 Introduction This post illustrates a key lemma that is used in proving the fundamental theorem of Galois theory (FTGT). Note that FTGT is not covered in this post. The focus of this post is on understanding and proving this lemma only. Here is the lemma from the book Galois Theory, 5th ed. by Stewart (2023): Lemma 12.1. Suppose that \( L/K \) is a field extension, \( M \) is an intermediate field, and \( \tau \) is a \( K \)-automorphism of \( L. \) Then \( \tau M^* \tau^{-1} = \tau(M)^{*}. \) The notation \( M^* \) denotes the group of all \( M \)-automorphisms of \( L \) with composition as the group operation. Note that Stewart writes \( \tau(M)^{*} = \tau M^* \tau^{-1} \) while stating the lemma but I have reversed the LHS and RHS to maintain consistency with the equations that appear in the discussion below. To build intuition for this lemma, I'll first present an illustration, followed by a proof. The discussion below assumes famil ... Read full article.