Editing Adele ring (section)

===Finiteness of the class number of a number field===

In the previous section the fact that the class number of a number field is finite had been used. Here this statement can be proved:

:'''Theorem (finiteness of the class number of a number field).''' Let <math>K</math> be a number field. Then <math>|\operatorname{Cl}_K|<\infty.</math>

'''Proof.''' The map

:<math>\begin{cases} I_K^1 \to J_K \\ \left ((\alpha_v)_{v < \infty}, (\alpha_v)_{v | \infty} \right ) \mapsto \prod_{v<\infty} \mathfrak{p}_v^{v(\alpha_v)} \end{cases}</math>

is surjective and therefore <math>\operatorname{Cl}_K</math> is the continuous image of the compact set <math>I_K^1/K^{\times}.</math> Thus, <math>\operatorname{Cl}_K</math> is compact. In addition, it is discrete and so finite.

'''Remark.''' There is a similar result for the case of a global function field. In this case, the so-called divisor group is defined. It can be shown that the quotient of the set of all divisors of degree <math>0</math> by the set of the principal divisors is a finite group.<ref>For more information, see {{harvnb|Cassels|Fröhlich|1967|p=71}}.</ref>