Editing Algebraic number theory (section)

==Major results==
===Finiteness of the class group===
One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field ''K'' is finite. This is a consequence of [[Minkowski's bound|Minkowski's theorem]] since there are only finitely many [[Integral ideal]]s with norm less than a fixed positive integer<ref>{{Cite web|title=A Computational Introduction to Algebraic Number Theory|url=https://wstein.org/books/ant/ant.pdf|last=Stein}}</ref> <sup>page 78</sup>. The order of the class group is called the [[Class number (number theory)|class number]], and is often denoted by the letter ''h''.

===Dirichlet's unit theorem===
{{Main|Dirichlet's unit theorem}}
Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units ''O''<sup>×</sup> of the ring of integers ''O''. Specifically, it states that ''O''<sup>×</sup> is isomorphic to ''G'' × '''Z'''<sup>''r''</sup>, where ''G'' is the finite [[cyclic group]] consisting of all the roots of unity in ''O'', and ''r'' = ''r''<sub>1</sub>&nbsp;+&nbsp;''r''<sub>2</sub>&nbsp;−&nbsp;1 (where ''r''<sub>1</sub> (respectively, ''r''<sub>2</sub>) denotes the number of real embeddings (respectively, pairs of conjugate non-real embeddings) of ''K''). In other words, ''O''<sup>×</sup> is a [[finitely generated abelian group]] of [[Rank of an abelian group|rank]] ''r''<sub>1</sub>&nbsp;+&nbsp;''r''<sub>2</sub>&nbsp;−&nbsp;1 whose torsion consists of the roots of unity in ''O''.

===Reciprocity laws===
{{Main|Reciprocity law}}

In terms of the [[Legendre symbol]], the law of quadratic reciprocity for positive odd primes states
:<math> \left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}.</math>

A '''reciprocity law''' is a generalization of the [[law of quadratic reciprocity]].

There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a [[power residue symbol]] (''p''/''q'') generalizing the [[Legendre symbol|quadratic reciprocity symbol]], that describes when a [[prime number]] is an ''n''th power residue [[modular arithmetic|modulo]] another prime, and gave a relation between (''p''/''q'') and (''q''/''p''). Hilbert reformulated the reciprocity laws as saying that a product over ''p'' of Hilbert symbols (''a'',''b''/''p''), taking values in roots of unity, is equal to 1. [[Emil Artin|Artin]]'s reformulated [[Artin reciprocity law|reciprocity law]] states that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.

===Class number formula===
{{Main|Class number formula}}
The '''class number formula''' relates many important invariants of a [[number field]] to a special value of its Dedekind zeta function.