## Any Automorphism of the Field of Real Numbers Must be the Identity Map

## Problem 507

Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.

Add to solve laterProve that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.

Add to solve laterProve that every finite group having more than two elements has a nontrivial automorphism.

(*Michigan State University, Abstract Algebra Qualifying Exam*)

Read solution

Let $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$.

Let $\Aut(N)$ be the group of automorphisms of $G$.

Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime.

Then prove that $N$ is contained in the center of $G$.

Let $H$ be a subgroup of a group $G$. We call $H$ **characteristic** in $G$ if for any automorphism $\sigma\in \Aut(G)$ of $G$, we have $\sigma(H)=H$.

**(a)** Prove that if $\sigma(H) \subset H$ for all $\sigma \in \Aut(G)$, then $H$ is characteristic in $G$.

**(b)** Prove that the center $Z(G)$ of $G$ is characteristic in $G$.

Show that $\Q(\sqrt{2+\sqrt{2}})$ is a cyclic quartic field, that is, it is a Galois extension of degree $4$ with cyclic Galois group.

Add to solve laterLet $\Q$ be the field of rational numbers.

**(a)** Is the polynomial $f(x)=x^2-2$ separable over $\Q$?

**(b)** Find the Galois group of $f(x)$ over $\Q$.

Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism.

The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation

\[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),\]
where $a_i \in A, b_i \in B$ for $i=1, 2$.

Let $f: A \to A’$ and $g:B \to B’$ be group isomorphisms. Define $\phi’: B’\to \Aut(A’)$ by sending $b’ \in B’$ to $f\circ \phi(g^{-1}(b’))\circ f^{-1}$.

\[\require{AMScd}

\begin{CD}

B @>{\phi}>> \Aut(A)\\

@A{g^{-1}}AA @VV{\sigma_f}V \\

B’ @>{\phi’}>> \Aut(A’)

\end{CD}\]
Here $\sigma_f:\Aut(A) \to \Aut(A’)$ is defined by $ \alpha \in \Aut(A) \mapsto f\alpha f^{-1}\in \Aut(A’)$.

Then show that

\[A \rtimes_{\phi} B \cong A’ \rtimes_{\phi’} B’.\]

An isomorphism from a group $G$ to itself is called* *an* automorphism *of $G$.

The set of all automorphism is denoted by $\Aut(G)$.

A subgroup $H$ of a group $G$ is called ** characteristic** in $G$ if for any $\phi \in \Aut(G)$, we have $\phi(H)=H$. In words, this means that each automorphism of $G$ maps $H$ to itself.

Prove the followings.

**(a)** If $H$ is characteristic in $G$, then $H$ is a normal subgroup of $G$.

**(b) **If $H$ is the unique subgroup of $G$ of a given order, then $H$ is characteristic in $G$.

**(c)** Suppose that a subgroup $K$ is characteristic in a group $H$ and $H$ is a normal subgroup of $G$. Then $K$ is a normal subgroup in $G$.