Circle dividing body
Circular dividing bodies (also: cyclotomic bodies ) are objects of study in the mathematical sub-area of algebraic number theory . In some respects they are particularly simple generalizations of the field of rational numbers .
- Definition: Let it be a natural number . Then the -th body of a circle is the extension of the body , which arises through the adjunction of the set of all -th roots of unity .
properties
- Is a primitive -th root of unit, then the minimal polynomial of the -th circle division polynomial , therefore is
- In particular, the body degree is with Euler's φ-function .
- Two circle division bodies and with are exactly the same if is odd and applies.
- The adjunction of the -th roots of unity to yields with
- The extension is Galois . The Galois group is isomorphic to is a primitive -th root of unity, so one element corresponds to the through
- defined automorphism of
- The wholeness ring of is with any primitive -th root of unity
- In particular, the whole ring of is equal to the ring of whole Gaussian numbers , the whole ring of is equal to the ring of Eisenstein numbers . These two number fields are the only algebraic extensions of the rational numbers, which are both circular and quadratic extension fields .
- A prime number is exactly then branched into if a divisor of is. is fully decomposed if and only if applies.
- If there is a prime power and a primitive -th root of unity, it is branched into undivided and pure. The only prime ideal above is the main ideal generated by:
Kronecker-Weber's Theorem
The Kronecker-Weber theorem (after L. Kronecker and H. Weber ) says that every algebraic number field with an Abelian Galois group is contained in a circle division field. The maximum Abelian expansion of arises from the adjunction of all roots of unity.
literature
- Serge Lang : Cyclotomic Fields I and II (= Graduate Texts in Mathematics. 121). Combined 2nd edition. Springer, New York NY et al. 1990, ISBN 3-540-96671-4 .
- Lawrence C. Washington : Introduction to Cyclotomic Fields (= Graduate Texts in Mathematics. 83). Springer, Berlin et al. 1982, ISBN 3-540-90622-3 (2nd edition. Springer, New York et al. 1997, ISBN 0-387-94762-0 ).