= Unit rank of a number field
{title2=$r_1+r_2-1$}
= Unit rank
{synonym}
The unit rank of a <number field> $K$ is the rank of the free abelian part of the <unit group> of its <ring of integers>. If $K$ has $r_1$ real embeddings and $r_2$ conjugate pairs of complex embeddings, <Dirichlet's unit theorem> gives
$$
\mathcal O_K^\times\cong\mu(K)\times\mathbb Z^{r_1+r_2-1},
$$
where $\mu(K)$ is the finite group of <roots of unity> in $K$. Hence the unit rank is $r_1+r_2-1$.<Algebraic Number Theory by J. S. Milne (course notes)>[Chapter 5, Theorem 5.1]{ref}
The <rational numbers> and imaginary quadratic fields have rank zero; real quadratic fields have rank one. In $\mathbb Q(\sqrt2)$, the unit $1+\sqrt2$ has norm $-1$ and its powers produce infinitely many units. The logarithms of the absolute embedding values of a unit sum to zero when complex pairs are counted twice. This places their logarithmic images in a hyperplane of dimension $r_1+r_2-1$, explaining the geometric meaning of the rank.
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