Automorphism group of a field

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article gives a basic definition in the following area: Galois theory
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Definition

Symbol-free definition

The automorphism group of a field is defined as a group whose elements are all the automorphisms of the field, and where the group operation is composition of automorphisms. In other words, it gets a group structure as a subgroup of the group of all permutations of the field.

Definition with symbols

The automorphism group of a field $K$, denoted $\mathrm{A}\mathrm{u}\mathrm{t}(K)$, is a set whose elements are automorphisms $\sigma :K\to K$, and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of $\mathrm{S}\mathrm{y}\mathrm{m}(K)$, the group of all permutations on $K$.