Fundamental Theorem Preview

Abstract Algebra · Axiom Academy

LESSON Fundamental Theorem of Galois Theory The beautiful correspondence between subgroups of the Galois group and intermediate field extensions The Fundamental Theorem establishes two maps that are inverses of each other: The most surprising aspect: the correspondence reverses inclusions . When we go from subgroups to fields, the containment order flips! Why does this happen? A larger subgroup has more automorphisms, so it must fix fewer elements (more symmetries = fewer invariants). Conversely, a smaller field is fixed by a larger group of automorphisms. Let's see how the fixed field operation works in detail. For a subgroup H ≤ Gal(K/F), the fixed field is: These are precisely the elements that every automorphism in H leaves unchanged. 4. Normal Subgroups and Galois Extensions The theorem has a beautiful refinement: normal subgroups correspond to Galois intermediate extensions . This connects group theory structure with field theory structure. Moreover, when this happens, we get an isomorphism: We can visualize the entire correspondence using lattice diagrams. The subgroup lattice and the field lattice are dual to each other—one is the upside-down version of the other. Bijection: Perfect one-to-one correspondence Inclusion-Reversing: H₁ ⊆ H₂ ⟺ E₂ ⊆ E₁ Degree Formula: [K:K^H] = |H| and [K^H:F] = |Gal(K/F)|/|H| Normal Correspondence: H normal ⟺ K^H/F Galois Quotient Isomorphism: Gal(K/F)/H ≅ Gal(K^H/F)

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