Galois Extension Definition

Abstract Algebra · Axiom Academy

Understanding when field extensions have maximal symmetry: the perfect balance of normality and separability. Normal: Every irreducible polynomial in F[x] that has one root in K splits completely in K Separable: Every element of K has a minimal polynomial with distinct roots Think of normality as "closure under conjugates" — if you have one root of a polynomial, you must have all of them. Separability ensures no repeated roots, which is automatic in characteristic 0. The most powerful characterization of Galois extensions comes from counting automorphisms: For any finite extension, we always have |Gal( K/F )| ≤ [ K:F ]. Equality holds precisely when the extension is Galois — when there are "enough" automorphisms to capture all the symmetry. 3. Splitting Fields Are Galois The most common way to construct Galois extensions is through splitting fields: Why this works: A splitting field is minimal — it contains all roots of f but nothing extra. This guarantees normality. If f is separable (distinct roots), the extension is separable. Let's verify that ℚ(√2, √3)/ℚ is Galois: Now let's see why ℚ(∛2)/ℚ is not Galois:

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