Algebraic number field: Difference between revisions
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==Unit group== | ==Unit group== | ||
The [[unit group]] ''U'' of the maximal order ''O''<sub>''K''</sub> is described by '''Dirichet's unit theorem''': | |||
''U'' is a finitely generated [[abelian group]] with [[free rank]] ''r''+''s''-1 and torsion subgroup the [[root of unity|roots of unity]] in ''K''. A free generator of ''U'' is termed a ''fundamental unit''. | |||
We let σ<sub>1</sub>,...,σ<sub>''r''+''s''</sub> 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'''<sup>''r''</sup>×'''C'''<sup>''s''</sup>. | |||
The ''logarithmic embedding'' Λ derived from Σ is defined by taking λ<sub>''i''</sub>(''x'') = log |σ<sub>i</sub>(x)| and is a map from ''K''* to '''R'''<sup>''r''+''s''</sup>: 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== | ==Splitting of primes== | ||
Revision as of 04:14, 1 January 2009
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.
Ring of integers
Unit group
The unit group U of the maximal order OK 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,...,σr+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 Rr×Cs.
The logarithmic embedding Λ derived from Σ is defined by taking λi(x) = log |σi(x)| and is a map from K* to Rr+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
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.