Dirichlet's unit theorem
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are.
The statement is that the group of units is finitely generated and has rank (maximal number of multiplicatively independent elements) equal to
where r1 is the number of real embeddings and r2 the number of conjugate pairs of complex embeddings of K. This characterisation of r1 and r2 is based on the idea that there will be as many ways to embed K in the complex number field as the degree n = [K : ℚ]; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that
Note that if K is Galois over ℚ then either r1 is non-zero or r2 is non-zero, but not both.
Other ways of determining r1 and r2 are
As an example, if K is a quadratic field, the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field. The theory for real quadratic fields is essentially the theory of Pell's equation.
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