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.

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.

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.

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 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.

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. 
• I.N. Stewart; D.O. Tall (1979). Algebraic number theory. Chapman and Hall. ISBN 0-412-13840-9.