In algebraic number theory, Dirichlet's unit theorem determines the rank of the group of units in the ring O_K of algebraic integers of a number field K. The statement is that the rank is In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space one would need to contain it; or alternatively how large a free abelian group it can contain as a subgroup. ... The word unit means any of several things: Physical unit, a fundamental quantity of measurement in science or engineering Units (computer program), a popular program that does unit conversion. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...

r + s − 1

where r is the number of real embeddings and 2s the number of complex embeddings of K. This characterisation of r and s 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 In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

n = r + 2s.

Other ways of determining r and s are

• use the primitive element to write K = Q(α), and then r is the number of conjugates of α that are real, 2s the number that are complex;
• write the tensor product of fields K ⊗_QR as a product of fields, there being r copies of R and s copies of C.

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. In mathematics, a primitive element for an extension of fields L/K is an element ζ of L such that L = K(ζ), or in other words such that L is generated by ζ over K. This means that every element of L can be written as a quotient of... In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the (other) roots of the minimal polynomial PK,α(t) of α over K. If K is given inside an algebraically closed field C, then the conjugates can be taken inside... In mathematics, the theory of fields in abstract algebra lacks a direct product: the direct product of two fields, considered as ring (mathematics) is never itself a field. ... In mathematics, a quadratic field is a field extension K/Q of the form where d is a non-zero rational number. ... Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

The rank is > 0 for all number fields other than Q and imaginary quadratic fields. The 'size' of the units is measured in general by a determinant called the regulator. In principle a basis for the units can be effectively computed; in practice the calculations are quite involved when n is large. In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

The torsion in the group of units is always a cyclic group generated by some root of unity. For a totally real field the torsion must therefore be only {1,−1}. In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In number theory , a number field K is called totally real if for each embedding of K into the complex numbers the image lies inside the real numbers. ...

There is a generalisation of the unit theorem to so-called S-units, determining the rank of the unit group in localizations of rings of integers. In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ...