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Abstract Algebra

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Field Extensions: Field Automorphism of K/F (definition)

ABSALG-55Q1NF

The set of automorphisms of a field $K$ (i.e. the bijective field homomorphisms from $K$ to $K$), denoted ${\rm Aut}(K)$, forms a group under composition of maps.

Let $K$ be a field extension of a field $F$. The group of automorphisms of $K/F$, also called the group of $F$-automorphisms of $K$, is given by the set:

${\rm Aut}_F(K):=\{\sigma\in{\rm Aut}(K)\mid \sigma(x)=x,\; {\rm all}\;x\in F\}$ under composition of maps

...and is a subgroup of ${\rm Aut}(K)$.

Which of the following are true?

Select ALL that apply.

A

${\rm Aut}_{\mathbb Q}({\mathbb Q}(\sqrt2))$ is isomorphic to ${\mathbb Z}/2{\mathbb Z}$

B

${\rm Aut}_{\mathbb Q}({\mathbb Q}(\sqrt[3]{2}))$, $\sqrt[3]{2}\in {\mathbb R}$, is isomorphic to ${\mathbb Z}/3{\mathbb Z}$

C

${\rm Aut}_{\mathbb Q}({\mathbb Q}(\sqrt[3]{2}))$, $\sqrt[3]{2}\in {\mathbb R}$, consists only of the identity map

D

${\rm Aut}_{\mathbb Q}({\mathbb Q}(\sqrt{-1},\sqrt[4]{2}))$, $\sqrt[4]{2}\in {\mathbb R}$, has order 4

E

If ${\mathbb Q}\subseteq F$ is a field and $\sigma\in{\rm Aut}(F)$ then $\sigma(q)=q$ for all $q\in{\mathbb Q}$