# Algebraic number field

In number theory, an **algebraic number field** is a principal object of study in algebraic number theory. The algebraic and arithmetic structure of a number field has applications in other areas of number theory, such as the resulotion of Diophantine equationss.

An *algebraic number field* *K* is a finite degree field extension of the field **Q** of rational numbers. The elements of *K* are thus algebraic numbers. Let *n* = [*K*:**Q**] be the degree of the extension.

## Real and complex embeddings

We may embed *K* into the algebraically closed field of complex numbers **C**. There are exactly *n* such embeddings: we can see this by taking α to be a primitive element for *K*/**Q**, and letting *f* be the minimal polynomial of α. Then the embeddings correspond to the *n* roots of *f* in **C**.
Some, say *r*, of these embeddings will actually have image in the real numbers, and the remaining embeddings will occur in complex conjugate pairs, say 2*s* such. We have *n*=*r*+2*s*.

We let σ_{1},...,σ_{r+s} denote a set of complex embeddings of *K* into **C**, with the proviso that we choose just one out of each complex conjugate pair. We can regard these as defining an embedding Σ of *K* into **R**^{r}×**C**^{s}. The map Σ is a group homomorphism on the additive group *K*^{+}.

## Ring of integers

The algebraic integers in a number field *K* form a subring denoted by *O*_{K}. This may be seen as the integral closure of the ring of integers **Z** in *K*. The ring of integers is an order, this is, a ring which is finitely generated as a **Z**-module and it is maximal with respect to this property, hence often called the *maximal order* of *K*.

The ring of integers is an integral domain, but does not in general have the desirable factorisation properties of the ring **Z**. For example, in the quadratic field generated by the rationals
and , the number can be factorised both as and ; all of , , and
are irreducible elements.

The ring of integers is a Dedekind domain, having unique factorisation of ideals into prime ideals.

### Fractional ideal

A **fractional ideal** of *K* is an *O*_{K}-submodule of *K*.

### Ideal class group

The fractional ideals of *K* form an abelian group under ideal multiplication with the fractional ideal *O*_{K} = *O*_{K}.1 as identity element. The principal ideals, fractional ideals of the form *O*_{K}.*x* for *x* in *K*, form a subgroup. Two fractional ideals are said to be in the same *ideal class* if one is a multiple of the other by some principal ideal. The quotient group is the **ideal class group**, denoted *H*(*K*). Hermite's theorem states that this group is *finite*. Its order *h*(*K*) is the *class number* of *K*.

A field has class number one if and only if its ring of integers is a principal ideal domain.

## Unit group

The unit group *U* of the maximal order *O*_{K} is described by **Dirichet's unit theorem**:
*U* is a finitely generated abelian group with free rank *r*+*s*-1 and torsion subgroup the roots of unity in *K*. A free generator of *U* is termed a *fundamental unit*.

The *logarithmic embedding* Λ derived from Σ is defined by taking λ_{i}(*x*) = log |σ_{i}(x)| and is a map from *K** to **R**^{r+s}: it is a group homomorphism. The Unit Theorem implies that this map has the roots of unity as kernel and maps *U* to a lattice of full rank in a hyperplane.

The **regulator** of *K* is the determinant of the lattice which is the image of *U* under Λ.

## Splitting of primes

## Zeta function

The *Dedekind zeta function* of the field *K* is a meromorphic function, defined for complex numbers *s* with real part satisfying by the Dirichlet series

where the sum extends over the set of integral ideals of *K*, and denotes their absolute norm.

This series is absolutely convergent on compact subsets of the half-plane . It thus defines a holomorphic function on this half-plane, and this can be extended by analytic continuation to a meromorphic function on the whole complex plane. It is holomorphic everywhere except at *s* = 1, where it has a simple pole.

The Dedekind zeta function has an *Euler product*:

where runs over prime ideals of the ring of integers, which formally expresses the unique factorisation of ideals of *O*_{K} into prime ideals.

## See also

- Cyclotomic field [r]: An algebraic number field generated over the rational numbers by roots of unity.
^{[e]} - Quadratic field [r]: A field which is an extension of its prime field of degree two.
^{[e]}

## References

- J.W.S. Cassels; A. Fröhlich (1967).
*Algebraic Number Theory)*. Academic Press. ISBN 012268950X. - A. Fröhlich; M.J. Taylor (1991).
*Algebraic number theory*. Cambridge University Press. ISBN 0-521-36664-X. - Gerald Janusz (1973).
*Algebraic Number Fields*. Academic Press. ISBN 0-12-380520-4. - Serge Lang (1986).
*Algebraic number theory*. Springer-Verlag. ISBN 0-387-94225-4. - P.J. McCarthy (1991).
*Algebraic extensions of fields*. Dover Publications. ISBN 0-486-66651-4. - W. Narkiewicz (1990).
*Elementary and analytic theory of algebraic numbers*, 2nd ed. Springer-Verlag/Polish Scientific Publishers PWN. ISBN 3-540-51250-0. - Pierre Samuel (1972).
*Algebraic number theory*. Hermann/Kershaw. - I.N. Stewart; D.O. Tall (1979).
*Algebraic number theory*. Chapman and Hall. ISBN 0-412-13840-9.