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 2s such. We have n=r+2s.

We let σ₁,...,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 $\scriptstyle {\sqrt {-5}}$, the number $6$ can be factorised both as $\scriptstyle 6=2\cdot 3$ and $\scriptstyle 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})$; all of $2$, $3$, $\scriptstyle 1+{\sqrt {-5}}$ and $\scriptstyle 1-{\sqrt {-5}}$ 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.

Zeta function

The Dedekind zeta function of the field K is a meromorphic function, defined for complex numbers s with real part satisfying $\Re (s)>1$ by the Dirichlet series

$\zeta _{k}(s)=\sum _{\mathfrak {a}}{\mathfrak {N}}({\mathfrak {a}})^{-s},$

where the sum extends over the set of integral ideals of K, and ${\mathfrak {N}}({\mathfrak {a}})$ denotes their absolute norm.

This series is absolutely convergent on compact subsets of the half-plane $\Re (s)>1$. 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:

$\zeta _{k}(s)=\sum _{\mathfrak {a}}{\mathfrak {N}}({\mathfrak {a}})^{-s}=\prod _{\mathfrak {p}}\left(1-{\mathfrak {N}}({\mathfrak {p}})^{-s}\right)^{-1},$

where ${\mathfrak {p}}$ 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.