# Galois Theory

**Type:** Concept  
**Domain:** Algebra  
**Codex URL:** /mathematics/algebra/galois-theory/  
**Entry status:** Live — v1.0 (2026-05-27)

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## Summary

Connects field extensions to group theory via the Galois group. A polynomial is solvable by radicals if and only if its Galois group is solvable. Introduced by Évariste Galois (1830–1832), published posthumously by Liouville (1846).

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## Fundamental Theorem of Galois Theory

For finite Galois extension K/F with G = Gal(K/F): subgroups H ≤ G correspond bijectively, order-reversingly, to intermediate fields. [K:E] = |H|, [E:F] = [G:H]. Normal subgroups correspond to Galois sub-extensions.

## Abel–Ruffini Theorem

No general radical formula exists for degree ≥ 5 polynomials, because Sₙ (n≥5) is not solvable (contains simple non-abelian Aₙ).

## Constructibility

Galois theory resolves:
- Doubling the cube (∛2) — impossible: [ℚ(∛2):ℚ]=3, not power of 2
- Trisecting an angle — impossible in general
- Squaring the circle — impossible: π transcendental (Lindemann, 1882)

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## Sources

### Tier 1
- Stewart, I. (2015). *Galois Theory*. 4th ed. CRC Press.
- Dummit, D.S. and Foote, R.M. (2004). *Abstract Algebra*. 3rd ed. Chapter 14.

### Tier 2
- Tignol, J-P. (2001). *Galois' Theory of Algebraic Equations*. World Scientific.

### Tier 3
- Livio, M. (2005). *The Equation That Couldn't Be Solved*. Simon & Schuster.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*