Galois Theory Intro: Concepts, Formulas and Examples

Galois theory studies the hidden symmetries inside polynomial equations. The two roots of x squared equals 2, square root of 2 and minus square root of 2, are mirror images of each other, and Galois theory makes that precise: it attaches to every polynomial a group of symmetries, the Galois group, that shuffles the roots in all ways respecting addition and multiplication.

This matters because it explains which equations have formulas. There are formulas for degrees 2, 3 and 4, but none for degree 5. Galois theory proves why: an equation is solvable by radicals exactly when its Galois group is a solvable group. The Abel-Ruffini theorem, no general radical formula for degree 5 and above, follows directly.

Key Concepts

• Field: a number system with addition, subtraction, multiplication and division, like the rationals Q or reals R.
• Field extension E/F: a bigger field containing a smaller one, for example Q(square root of 2) over Q.
• Degree [E : F]: the dimension of E as a vector space over F; [Q(square root of 2) : Q] = 2.
• Automorphism: a symmetry of a field preserving addition and multiplication.
• Galois group Gal(E/F): all automorphisms of E fixing every element of F.
• Splitting field: the smallest field where a polynomial factors completely into linear factors.

Formulas

• Order bound: |Gal(E/F)| is at most [E : F], with equality for Galois extensions.
• Fundamental theorem: for a Galois extension, intermediate fields correspond exactly to subgroups of the Galois group, reversing inclusion: bigger fields match smaller subgroups, with [E : K] = |H| and [K : F] = |G|/|H|.
• Galois criterion: a finite extension is Galois exactly when it is the splitting field of a separable polynomial (no repeated roots).

Worked Example

Take x squared minus 2 over Q. Its roots are r1 = square root of 2 (about 1.414) and r2 = minus square root of 2. The splitting field is E = Q(square root of 2), since r2 = minus r1 is already there. The degree [E : Q] = 2 with basis 1, square root of 2. An automorphism fixing Q is determined by where square root of 2 goes, and since its square 2 must be preserved, only square root of 2 or minus square root of 2 are possible. So Gal(E/Q) has exactly two elements: the identity and conjugation sigma(a + b square root of 2) = a minus b square root of 2. The order 2 equals the degree 2, confirming a Galois extension. Its subgroups, the trivial one and the whole group, correspond to the intermediate fields Q(square root of 2) and Q respectively.

Common Misconceptions

• Galois theory says quintics have no roots. False: every degree n polynomial has n complex roots. What is impossible is one general radical formula working for all degree 5 polynomials.
• The Galois group is all permutations of the roots. Only permutations achievable by field automorphisms count. For x to the 4th minus 2 the group has 8 elements, not all 24 in S4.
• It only matters for unsolvable equations. It also decides straightedge-and-compass constructions and describes finite fields used in codes.

Real-World Applications

• Algebra: the Abel-Ruffini theorem and the classification of radical-solvable polynomials.
• Geometry: which regular polygons are constructible, such as Gauss's 17-gon.
• Codes and cryptography: finite-field Galois theory underlies Reed-Solomon codes in QR codes and the AES cipher.

Galois theory converts solvability questions into group theory via the fundamental theorem's field-subgroup dictionary. The x squared minus 2 example holds every key idea in miniature: splitting field, degree, automorphisms and the correspondence.