Dirichlet's unit theorem

Named after Peter Gustav Lejeune Dirichlet Dirichlet unit theorem is one of the first results of algebraic number theory. The theorem describes the structure of the unit group of wholeness ring of an algebraic number field.

Formulation

It is an algebraic number field and his wholeness ring. Then the unit group is finitely generated, and the rank of its free fraction is equal to

Here, the number of inclusions and the number of pairs of complex -conjugated inclusions ( inclusions are no real ). If the extension is Galois, then or equal.

The Torsionsanteil the unit group is the group of roots of unity in.

Sketch of proof in a special case

Let ( so we choose already a real embedding ); then, and the unit group

( The equation is named pellsche equation. )

In this case, and the Dirichlet unit theorem states thus requires that the rank of is equal to 1.

For example, since is a unit that is not a root of unity, the rank at least 1 must be. If the rank is larger, could not be a discrete subgroup of, and you know that a subgroup of is either discrete or dense. There would be a unit that is "approximately" 1. Now, however, and two numbers whose product is, so is the one of them about 1, so is the other about. On the other hand, they differ by the number that is "essential" greater than the distance between and, where is. But if, as is obvious, so we only get the roots of unity.

• Algebra
• Number Theory
• Algebraic Number Theory
• Set ( mathematics)