TITLE: Units in ring of integers are exactly those with norm {-1,1}
QUESTION [5 upvotes]: Let $K$ be an algebraic number field and $R$ be the ring of the integers of $K$. Show that an element $u\in R$ is a unit of $R$ if and only if $N_{K/\mathbb{Q}}(u)\in \{-1,1\}$.

It is easy to show units are of norm {-1,1}. But on the other hand, I have no idea.

REPLY [7 votes]: Without using ideals:  - if $u\in R$ is a unit, then  $N(u)N(u^{-1}) = 1$, an equation in $\mathbf Z$, hence $N(u) =\pm 1$  - conversely, if $u\in R$ has norm $\pm1$, as an algebraic integer $u$ is a root of a polynomial of the form  $X^n + ... + a_1 X\pm1\in \mathbf Z[X]$, hence $\pm(u^{n-1} + ... + a_1)\in R$ is the inverse of $u$ .