When Galois Groups are Largest

Abstract Algebra · Axiom Academy

INTRO When Are Galois Groups Largest? Explore the relationship between extension degree and Galois group size. When do they match? For a field extension , we can measure two things: the degree and the size of the Galois group . Click on different extensions to compare these values. Let's look at more examples. Click each card to reveal its properties and the minimal polynomial. Drag each extension to the correct category based on whether or . Pay attention to what makes them different! Look at how the minimal polynomials factor in each extension. The pattern becomes clear! Understanding Galois Extensions When every element's minimal polynomial splits completely in the extension, we call it a Galois extension . These extensions have maximal symmetry: the Galois group captures all possible automorphisms. When minimal polynomials have irreducible factors of degree greater than 1 in the extension, we get fewer automorphisms than the extension degree. The extension lacks complete symmetry. For any field extension: Equality holds (making it Galois) if and only if: 1. The extension is normal (all minimal polynomials split) 2. The extension is separable (no repeated roots) Over fields of characteristic 0 (like ), separability is automatic, so normality is the key!

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