Units of $\mathbb Z[X]/(X^n+1)$?

Question

What are the units of the cyclotomic ring $\mathbb Z[X]/(X^n+1)$, with $n$ being a power of $2$?

I am starting to think that the set $\{\pm X^k,k=0,\dots,n-1\}$ contains all units, is that so ?

Answer 1

It is not true. Dirichlet's theorem says that the unit group of a number field $K$ is finitely generated of rank $r_1+r_2-1$ where $r_1$ and $r_2$ are the number of real (resp. half the number of complex) embeddings of $K$.

The field $\mathbb Q(\zeta)$, $\zeta^n+1=0$, with $n>1$ a power of $2$, has no real embeddings and it has exactly $n$ complex embeddings. Thus the group of units of its ring of integers (which is $\mathbb Z[\zeta]$) has rank $n/2-1$. In particular, it's not a finite group for $n>2$.

What is true is that you described entirely the torsion in the group of units. But there are lots of other units generally.