Editing Algebraic number theory (section)

===Units===
The integers have only two units, {{math|1}} and {{math|&minus;1}}. Other rings of integers may admit more units. The Gaussian integers have four units, the previous two as well as {{math|±''i''}}. The [[Eisenstein integers]] {{math|'''Z'''[exp(2&pi;''i'' / 3)]}} have six units. The integers in real quadratic number fields have infinitely many units. For example, in {{math|'''Z'''[√{{Overline|3}}]}}, every power of {{math|2 + √{{Overline|3}}}} is a unit, and all these powers are distinct.

In general, the group of units of {{math|''O''}}, denoted {{math|''O''<sup>×</sup>}}, is a finitely generated abelian group. The [[fundamental theorem of finitely generated abelian groups]] therefore implies that it is a direct sum of a torsion part and a free part. Reinterpreting this in the context of a number field, the torsion part consists of the [[root of unity|roots of unity]] that lie in {{math|''O''}}. This group is cyclic. The free part is described by [[Dirichlet's unit theorem]]. This theorem says that rank of the free part is {{math|''r''<sub>1</sub> + ''r''<sub>2</sub> &minus; 1}}. Thus, for example, the only fields for which the rank of the free part is zero are {{math|'''Q'''}} and the imaginary quadratic fields. A more precise statement giving the structure of ''O''<sup>×</sup> ⊗<sub>'''Z'''</sub> '''Q''' as a [[Galois module]] for the Galois group of ''K''/'''Q''' is also possible.<ref>See proposition VIII.8.6.11 of {{harvnb|Neukirch|Schmidt|Wingberg|2000}}</ref>

The free part of the unit group can be studied using the infinite places of {{math|''K''}}. Consider the function

:<math>\begin{cases} L: K^\times \to \mathbf{R}^{r_1 + r_2} \\ L(x) = (\log |x|_v)_v \end{cases}</math>
 
where {{math|''v''}} varies over the infinite places of {{math|''K''}} and |·|<sub>''v''</sub> is the absolute value associated with {{math|''v''}}. The function {{math|''L''}} is a homomorphism from {{math|''K''<sup>×</sup>}} to a real vector space. It can be shown that the image of {{math|''O''<sup>×</sup>}} is a lattice that spans the hyperplane defined by <math>x_1 + \cdots + x_{r_1 + r_2} = 0.</math> The covolume of this lattice is the '''regulator''' of the number field. One of the simplifications made possible by working with the adele ring is that there is a single object, the [[idele class group]], that describes both the quotient by this lattice and the ideal class group.