Finding Automorphisms of Q(∛2)

Abstract Algebra · Axiom Academy

EXAMPLE Finding Automorphisms of Q(∛2) Systematically determine all field automorphisms and discover why this extension has only the identity automorphism. Excellent work! You've discovered why Q(∛2) has a trivial Galois group. Here's what we learned: Minimal Polynomial Constraint: Any automorphism σ must map ∛2 to another root of x³ - 2, since it preserves polynomial relations. Real vs. Complex: The polynomial x³ - 2 has three roots, but only ∛2 is real. The other two roots (ω∛2 and ω²∛2) are complex and not in Q(∛2). Only Identity Automorphism: Since σ(∛2) must be in Q(∛2) and the only root in this field is ∛2 itself, we have σ(∛2) = ∛2, making σ the identity. The Galois Group: Aut(Q(∛2)/Q) = id , so |Aut(Q(∛2)/Q)| = 1. This is a trivial Galois group. Key Insight: To get interesting (non-trivial) Galois groups, we need to include ALL roots of the minimal polynomial in our field extension! For x³ - 2, we'd need the splitting field Q(∛2, ω), which contains all three roots. This example illustrates why Galois theory focuses on splitting fields and why we need all conjugate roots to obtain the full symmetry group. The splitting field of x³ - 2 has Galois group S₃ (order 6), much more interesting than our trivial group!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.