An automorphism: a symmetry of a field
An automorphism of a field K is a relabeling of its elements that preserves all the arithmetic: σ(a + b) = σ(a) + σ(b) and σ(ab) = σ(a)σ(b). For an extension K/F we add one rule — σ must fix every element of the base field F, leaving the rationals untouched.
The classic example: complex conjugation. The map a + bi ↦ a − bi is an automorphism of C that fixes every real number. It is exactly a conjugate symmetry — and it swaps the two roots i and −i of x² + 1.
The Galois group
Collect every automorphism of K that fixes F. They form a group under composition — call it the Galois group Gal(K/F). The identity is the do-nothing map, the inverse of a relabeling undoes it, and composing two symmetries is again a symmetry. Everything you learned about groups now applies.
Because each σ just shuffles the roots, the Galois group acts as a group of permutations of the roots — a subgroup of the full symmetric group on n roots. Often it is a proper subgroup, not the whole thing, because the roots may have hidden relations the symmetries must respect.
Working one out by hand
Gal(Q(√2, √3) / Q): Roots to shuffle: √2 → ±√2, √3 → ±√3 (independently) Each automorphism is determined by these two sign choices: e : √2 → √2, √3 → √3 (identity) σ : √2 → −√2, √3 → √3 τ : √2 → √2, √3 → −√3 στ : √2 → −√2, √3 → −√3 4 elements, every one its own inverse ⇒ the Klein four-group. Note |Gal| = 4 = [Q(√2,√3) : Q]. The order MATCHES the degree.
That last line — |Gal(K/F)| equals the degree [K : F] — is no accident. It holds exactly when K is a “nice” extension (a splitting field of a separable polynomial). When the sizes match like this, the extension is called Galois, and we are ready for the theorem that ties the two worlds together.