Units are algebraic integers whose multiplicative inverse is also integral. In number fields they form a finitely generated abelian group whose rank is determined solely by the real and complex embeddings.
Norm criterion
Proposition. For \(u\in\mathcal O_K\), \[ u\in\mathcal O_K^\times \quad\Longleftrightarrow\quad N_{K/\mathbb Q}(u)=\pm1. \]
Proof. If \(uv=1\) with \(v\in\mathcal O_K\), multiplicativity gives \(N(u)N(v)=1\), and both norms are integers. Conversely, the characteristic polynomial of multiplication by \(u\) has constant term \(\pm N(u)=\pm1\); rearranging its monic equation expresses \(u^{-1}\) as an integer polynomial in \(u\), hence as an algebraic integer.
The logarithmic embedding
Let \(K\) have \(r_1\) real embeddings and \(r_2\) pairs of complex embeddings. Define \[ L:K^\times\to\mathbb R^{r_1+r_2},\qquad L(\alpha)= \bigl(\log|\sigma_1\alpha|,\ldots, \log|\sigma_{r_1}\alpha|, 2\log|\tau_1\alpha|,\ldots,2\log|\tau_{r_2}\alpha|\bigr). \] For a unit \(u\), the product formula gives \[ \sum_iL_i(u)=\log|N(u)|=0. \] Thus units map into the hyperplane \[ H=\{x:\sum_ix_i=0\}. \]
Theorem (Dirichlet unit theorem). \[ \mathcal O_K^\times\cong \mu(K)\times\mathbb Z^{r_1+r_2-1}, \] where \(\mu(K)\) is the finite cyclic group of roots of unity in \(K\). Equivalently, \(L(\mathcal O_K^\times)\) is a full lattice in \(H\).
Proof outline. The kernel of \(L\) consists of algebraic integers all of whose conjugates have absolute value \(1\). Bounded coefficients of their minimal polynomials show that only finitely many exist; Kronecker's theorem identifies them as roots of unity. Discreteness of the image follows by the same bounded-conjugate argument. Minkowski's convex-body theorem produces enough independent units to make the image span the \((r_1+r_2-1)\)-dimensional hyperplane. A discrete full-rank subgroup is a lattice, yielding the decomposition.
Real quadratic fields and Pell equations
Example. In \(\mathbb Q(\sqrt d)\), \(d>0\) squarefree, the unit rank is one. A fundamental unit \(\varepsilon>1\) generates all units up to sign: \[ \mathcal O_K^\times=\{\pm\varepsilon^n:n\in\mathbb Z\}. \] For \(\mathbb Z[\sqrt d]\), units of norm \(1\) correspond to solutions of \[ x^2-dy^2=1. \] Continued fractions of \(\sqrt d\) construct the fundamental solution.
The regulator
Choose fundamental units \(\varepsilon_1,\ldots,\varepsilon_r\), where \(r=r_1+r_2-1\). The covolume of their logarithmic lattice is the regulator \(R_K\). Together with the class number, roots of unity, and discriminant, it appears in the residue formula \[ \operatorname*{Res}_{s=1}\zeta_K(s) =\frac{2^{r_1}(2\pi)^{r_2}h_KR_K} {|\mu(K)|\sqrt{|d_K|}}. \] This analytic class-number formula unites ideals, units, geometry of numbers, and the Dedekind zeta function.