Finding All Subfields of Q(ω₈)

Abstract Algebra · Axiom Academy

EXAMPLE Finding All Subfields of Q(ω₈) Use Galois correspondence to systematically find all intermediate fields in the extension Q(ω₈)/Q Excellent work! You've successfully found all subfields of Q(ω₈) using Galois correspondence. Here's what we learned: Galois Group Structure: Gal(Q(ω₈)/Q) ≅ (Z/8Z)* ≅ Z₂ × Z₂, which has exactly 5 subgroups (including trivial and whole group) Fundamental Theorem: There's a one-to-one correspondence between subgroups of Gal(Q(ω₈)/Q) and intermediate fields between Q and Q(ω₈) The Three Intermediate Fields: Q(i), Q(√2), and Q(i√2) are the three proper intermediate fields, each of degree 2 over Q Degree Formula: If H is a subgroup, then [Q(ω₈)ᴴ : Q] = |H| and [Q(ω₈) : Q(ω₈)ᴴ] = |G/H| Tower Structure: The extension can be broken down: Q ⊂ Q(i) ⊂ Q(ω₈), or Q ⊂ Q(√2) ⊂ Q(ω₈), or Q ⊂ Q(i√2) ⊂ Q(ω₈) This technique of using Galois correspondence is fundamental in field theory and generalizes to finding subfields of any Galois extension. The key is understanding the group structure and applying the correspondence systematically!

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