automorphism

An automorphism is a symmetry of a mathematical structure: a way of shuffling its elements around that leaves the structure looking exactly the same as before. Like rotating a square so it lands back on itself, an automorphism relabels the elements yet preserves every operation and relation, so you could not tell from the inside that anything moved.

Precisely, an automorphism is an isomorphism from a structure to itself — a bijection (one-to-one and onto) that respects all the structure's operations. For a field, an automorphism is a bijection f with f(a + b) = f(a) + f(b) and f(a·b) = f(a)·f(b) for all a, b; it automatically fixes 0 and 1 and commutes with every algebraic operation.

The set of all automorphisms of a structure forms a group under composition: composing two symmetries gives another symmetry, the identity map is a symmetry, and every automorphism can be undone. In Galois theory we focus on field automorphisms that fix the base field, and that group of symmetries is exactly what unlocks the theory.