A Key Lemma in Proving the Fundamental Theorem of Galois Theory
2025-03-15
This blog post proves a key lemma used in proving the Fundamental Theorem of Galois Theory (FTGT). Lemma 12.1 states: If L/K is a field extension, M is an intermediate field, and τ is a K-automorphism of L, then τM*τ⁻¹ = τ(M)*. The post uses a concrete example (L = Q(√2, √3), K = Q, M = Q(√2)) to illustrate the lemma and provides a complete proof, showing both τM*τ⁻¹ ⊆ τ(M)* and τM*τ⁻¹ ⊇ τ(M)*. This is crucial for understanding Galois theory.