The ring $R$ of algebraic integers of any algebraic number field contains a finite set $H = \{\eta_1,\, \eta_2,\, \ldots,\, \eta_t\}$ of so-called fundamental units such that every unit $\varepsilon$ of $R$ is a power product of these, multiplied by a root of unity: $$\varepsilon = \zeta\!\cdot\!\eta_1^{k_1}\eta_2^{k_2}\ldots\eta_t^{k_t}$$ Conversely, every such element $\varepsilon$ of the field is a unit of $R$

Examples: units of quadratic fields, units of certain cubic fields

For some algebraic number fields, such as all imaginary quadratic fields, the set $H$ may be empty ($t = 0$ . In the case of a single fundamental unit ($t = 1$ , which occurs e.g. in all real quadratic fields, there are two alternative units $\eta$ and its conjugate $\overline{\eta}$ which one can use as fundamental unit; then we can speak of the uniquely determined fundamental unit $\eta_1$ which is greater than 1.