System of fundamental units

Assume the Axiom of Choice (The Axiom of Choice). Let $K$ be a number field of signature $(r_1,r_2)$ and unit rank $r=r_1+r_2-1$ (Dirichlet unit theorem). By the full-lattice theorem the image $\lambda ({\mathcal{O}}_K^{\times })$ is a free abelian subgroup of the hyperplane $H$ of rank $r$ (The logarithmic unit image is a full lattice, Free abelian group on a set). A system of fundamental units of $K$ is a tuple $({\epsilon }_1,\dots ,{\epsilon }_r)$ of units of ${\mathcal{O}}_K$ such that $\lambda ({\epsilon }_1),\dots ,\lambda ({\epsilon }_r)$ is a $\mathbb{Z}$-basis of $\lambda ({\mathcal{O}}_K^{\times })$, that is

$\lambda ({\mathcal{O}}_K^{\times })=\mathbb{Z}\lambda ({\epsilon }_1)\oplus \cdots \oplus \mathbb{Z}\lambda ({\epsilon }_r)\mathrm{.}$

Existence. The unit theorem gives ${\mathcal{O}}_K^{\times }\cong \mu (K)\times {\mathbb{Z}}^r$ (Dirichlet unit theorem), so $\lambda ({\mathcal{O}}_K^{\times })$ is a free abelian group of rank $r$ and has a $\mathbb{Z}$-basis; since $\lambda$ maps ${\mathcal{O}}_K^{\times }$ onto its image, each basis vector is $\lambda ({\epsilon }_i)$ for some unit ${\epsilon }_i$. This exhibits a system of fundamental units, and the construction makes only the finitely many choices of preimages of a finite basis, so the Axiom of Choice is used here only through the unit theorem. In rank $r=0$ the empty tuple is the unique system of fundamental units.

The equivalent product description. A tuple $({\epsilon }_1,\dots ,{\epsilon }_r)$ is a system of fundamental units if and only if every unit $u\in {\mathcal{O}}_K^{\times }$ admits a unique expression

$u=\zeta {\epsilon }_1^{m_1}\cdots {\epsilon }_r^{m_r},\zeta \in \mu (K),m_1,\dots ,m_r\in \mathbb{Z}\mathrm{.}$

Indeed, if the $\lambda ({\epsilon }_i)$ form a $\mathbb{Z}$-basis and $u\in {\mathcal{O}}_K^{\times }$, then $\lambda (u)={\sum }_i{m}_i\lambda ({\epsilon }_i)=\lambda ({\epsilon }_1^{m_1}\cdots {\epsilon }_r^{m_r})$ for unique integers $m_i$, so $u({\epsilon }_1^{m_1}\cdots {\epsilon }_r^{m_r}{)}^{-1}$ lies in the kernel of $\lambda$ on ${\mathcal{O}}_K^{\times }$, which is $\mu (K)$ (Kernel of the unit logarithm is the roots of unity); uniqueness of the exponents follows from the $\mathbb{Z}$-independence of the basis and then uniqueness of $\zeta$ from cancellation in the group ${\mathcal{O}}_K^{\times }$ (The units of a ring are the invertible elements of its multiplicative monoid, and $R^{\times }$ is a group under multiplication; $0\in R^{\times }$ only in the zero ring). Conversely, if every unit has such a unique expression, then ${\sum }_i{m}_i\lambda ({\epsilon }_i)=0$ forces the unit ${\epsilon }_1^{m_1}\cdots {\epsilon }_r^{m_r}$ to lie in $\mu (K)$, hence by uniqueness all $m_i=0$, so the $\lambda ({\epsilon }_i)$ are $\mathbb{Z}$-independent; and applying $\lambda$ to the expression of an arbitrary unit shows that they generate $\lambda ({\mathcal{O}}_K^{\times })$. Thus the two descriptions of the definition agree.

A system is auxiliary data, not canonical field data. Different systems of fundamental units are related by a unimodular integer change of coordinates: the tuples $(\lambda ({\epsilon }_i))$ and $(\lambda ({\epsilon }_i^{\mathrm{\prime }}))$ are two $\mathbb{Z}$-bases of the same free abelian group, so $\lambda ({\epsilon }_i^{\mathrm{\prime }})={\sum }_j{c}_{ij}\lambda ({\epsilon }_j)$ with $(c_{ij})\in {\mathrm{GL}}_r(\mathbb{Z})$. No system is singled out by the field, and the definition introduces no sign or ordering convention: the regulator constructed from these units is independent of the system, a fact proved separately.