If $n$ is a square-free integer such that $n >1$, and $ K = \mathbb Q ( \sqrt n )$. Let $ A_K$ the ring of algebraic integers. Show that $ A_K ^ \times $ is a finitely generated abelian group and compute its rank. (Here $ A_K^\times$ denotes the group of units of $A_K$).
I have no idea how to get started. Any help is appreciated.
Ok, let me give it a try.
$i)$ If $n \not\equiv 1 \pmod 4$, then the ring of integers is $\mathbb Z[\sqrt n]$.
$ii)$ An element $a+b\sqrt n$ in the ring of integers is a unit if and only if $a^2-nb^2=1$.
$iii)$ The equation $a^2-nb^2=1$ is called Pell's Equation and is very famous, and is known to have one fundamental integer solution of which all other integer solutions are powers of (upto a sign).
Try to fill in the gaps? I am not sure how much you know. The $n\equiv 1 \pmod 4$ case might be a little more thorny.