Fixed Fields

Abstract Algebra · Axiom Academy

LESSON Fixed Fields in Galois Theory Understanding the correspondence between subgroups of the Galois group and intermediate field extensions Think of automorphisms in H as "symmetries" of the field K . The fixed field K^H contains exactly those elements that are "symmetric" with respect to all these symmetries. There's a fundamental inverse relationship between subgroups and fixed fields: Why? If H_2 contains more automorphisms than H_1 , then an element must be fixed by more automorphisms to be in K^ H_2 . This is a stricter requirement, so fewer elements qualify! Let's examine the splitting field of (x^2-2)(x^2-3) over . This is , which has degree 4 over . This group has 5 subgroups (including the trivial ones), and each corresponds to an intermediate field between and K . 4. The Complete Correspondence The Fundamental Theorem of Galois Theory tells us there's a one-to-one, order-reversing correspondence between: (trivial subgroup) ↔ K (entire field) G (entire group) ↔ (base field)

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