What is the meaning of "fix" in field theory?

Question

What is the meaning of "fix" in field theory?

Example: I found a definition of field automorphism,

A field automorphism fixes the smallest field containing $1$, which is $\Bbb Q$, the rational numbers, in the case of field characteristic zero.

The set of automorphisms of $F$ which fix a smaller field $F'$ forms a group, by composition, called the Galois group, written $\operatorname{Gal}(F/F')$. For example, take $F'=\Bbb Q$, the rational numbers, and ...

Answer 1

Let's get concrete for a moment. Consider the field automorphism:

$$f : \mathbb{C} \rightarrow \mathbb{C}\,\,\hbox{where}\,\,f(a + ib) = a - ib$$

This is commonly known as complex conjugation. (It's obvious that it's an invertible and that the domain and codomain are the same, and you can verify for yourself that it is a homomorphism if you want.)

If $z \in \mathbb{R}$, then $f(z) = z$. $\mathbb{R}$ is a subfield of $\mathbb{C}$, so we say that this subfield is "fixed" by the automorphism.

Answer 2

Basically, if $K$ is a field, $f: K \to K$ is a field automorphism then $f$ fixes some subfield $F$ of $K$ iff

$$ f|_{F} = id_F $$

i.e.

$$ \forall x \in F, f(x) = x $$

Answer 3

I disagree with the other two answers. They tell you what the author meant, however this is not necessarily the meaning of "fixes" in mathematics.

The author is imprecise. If an automorphism $\sigma$ fixes a subfield $S$ it could mean two things.

• It fixes the elements of the subfield, $\sigma(s)=s$ for all $s\in S$. That is, it fixes the subfield pointwise. So $\sigma$ induces the trivial automorphism of $S$.

• It fixes the subfield, but moves the elements around within the subfield, $\sigma(S)=S$. That is, it fixes the subfield setwise. So $\sigma$ induces an automorphism of $S$, but not necessarily the trivial automorphism.

For example, in group theory an inner automorphism will fix all normal subgroups setwise but not necessarily pointwise. For an explicit example, think of the semidirect product of $H=\mathbb{Z}_2\times\mathbb{Z}_2$ with $K=\mathbb{Z}_2$ where the action of the non-trivial element of $\mathbb{Z}_2$ on $H$ swaps the two copies of $\mathbb{Z}_2$. This fixes $H$ setwise but not pointwise, although it fixes the "third" copy of $\mathbb{Z}_2$ in $H$ pointwise. (If you don't know about semidirect products, then forget about normal subgroups and just consider $\mathbb{Z}_2\times\mathbb{Z}_2$ with the same automorphism.)