How can I solve this? Automorphism of $\mathbb Q(\sqrt{5})/ \mathbb Q)$ How can I solve $\text{Aut} (\mathbb Q(\sqrt{5})/ \mathbb Q)$ ? What does this look like? I appreciate any help. Thanks in advance!
 A: An automorphism must take a root of $x^2 - 5$ to another root. Thus $\sqrt{5} \mapsto \pm \sqrt{5}$. For any ring homomorphism, $1 \mapsto 1$. Since $1, \sqrt{5}$ is a generating set, this classifies all the automorphisms.
A: $Aut( \mathbb{Q}(\sqrt{5})$/ $ \mathbb{Q})=\{\sigma,id\}$ 
where $\sigma(\sqrt{5})=-\sqrt{5}$ and $\sigma(q)=q, \forall q \in \mathbb{Q}$ 
Thus  $Aut( \mathbb{Q}(\sqrt{5})$/ $\mathbb{Q})$ is isomorphic with $\mathbb{Z}_2$.
A: If you know that $\sqrt{5} \not\in \Bbb Q$, it follows that $x^2 - 5$ is irreducible in $\Bbb Q[x]$. Indeed, if $\alpha$ is any root of $x^2 - 5$ in an extension of $\Bbb Q$, it follows that in $\Bbb Q(\alpha)$ we have the factorization:
$x^2 - 5 = (x + \alpha)(x - \alpha)$.
Clearly, then, $\Bbb Q(\alpha)$ is a normal extension generated by the separable element $\alpha$. Even easier, since $\text{char}(\Bbb Q) = 0$, this is automatically a separable extension.
In short, $\Bbb Q(\alpha)$ is a Galois extension, and thus $\text{Aut}(\Bbb Q(\alpha)/\Bbb Q) = \text{Gal}(\Bbb Q(\alpha)/\Bbb Q)$, and we have:
$|\text{Aut}(\Bbb Q(\alpha)/\Bbb Q)| = [\Bbb Q(\alpha):\Bbb Q] = 2 = \text{deg}(x^2 - 5)$.
So the group of automorphisms (which in this case is the galois group) is cyclic of order $2$.
So this group has two elements: the identity automorphism, and an involution automorphism.
It is clear from the automorphism property of any element of this group, that it must send the set: $\{\alpha,-\alpha\}$ to itself, that is, that it is (isomorphic to) a subgroup of $S_2$ of order $2$. Since $|S_2| = 2$, the involution must send $\alpha \mapsto -\alpha$ (a transposition of the roots).
Explicitly, if we write (regarding $\Bbb Q(\sqrt{5})$ as a subfield of the complex numbers $\Bbb C$, for example):
$\Bbb Q(\sqrt{5}) = \{a + b\sqrt{5}: a,b \in \Bbb Q\}$
then the action of our two automorphisms are as follows:
$a+b\sqrt{5} \mapsto a+b\sqrt{5}$ (the identity automorphism)
$a+b\sqrt{5} \mapsto a-b\sqrt{5}$ (the involution, or conjugation map).
