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 . ${\displaystyle n>2}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} (\mu _{n})}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mu _{n}}$ ${\displaystyle n}$

properties

• Is a primitive -th root of unit, then the minimal polynomial of the -th circle division polynomial , therefore is ${\displaystyle \zeta _{n}}$ ${\displaystyle n}$ ${\displaystyle \zeta _{n}}$ ${\displaystyle n}$ ${\displaystyle \Phi _{n}}$

${\displaystyle \mathbb {Q} (\mu _{n})=\mathbb {Q} (\zeta _{n})\cong \mathbb {Q} [T]/(\Phi _{n}(T)).}$

In particular, the body degree is with Euler's φ-function . ${\displaystyle [\mathbb {Q} (\mu _{n}):\mathbb {Q} ]=\varphi (n)}$

• Two circle division bodies and with are exactly the same if is odd and applies. ${\displaystyle \mathbb {Q} (\mu _{n})}$ ${\displaystyle \mathbb {Q} \mathbb {(} \mu _{m})}$ ${\displaystyle n<m}$ ${\displaystyle n}$ ${\displaystyle m=2n}$

• The adjunction of the -th roots of unity to yields with ${\displaystyle m}$ ${\displaystyle \mathbb {Q} (\mu _{n})}$ ${\displaystyle \mathbb {Q} (\mu _{N})}$ ${\displaystyle N=\mathrm {kgV} (m,n).}$

• The extension is Galois . The Galois group is isomorphic to is a primitive -th root of unity, so one element corresponds to the through ${\displaystyle \mathbb {Q} (\mu _{n})|\mathbb {Q} }$ ${\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times };}$ ${\displaystyle \zeta _{n}}$ ${\displaystyle n}$ ${\displaystyle k\in (\mathbb {Z} /n\mathbb {Z} )^{\times }}$

${\displaystyle \zeta _{n}\mapsto \zeta _{n}^{k}}$

defined automorphism of ${\displaystyle \mathbb {Q} (\mu _{n}).}$

• The wholeness ring of is with any primitive -th root of unity ${\displaystyle \mathbb {Q} (\mu _{n})}$ ${\displaystyle \mathbb {Z} [\zeta _{n}]}$ ${\displaystyle n}$ ${\displaystyle \zeta _{n}.}$

• 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 . ${\displaystyle \mathbb {Q} (\mu _{4})=\mathbb {Q} ({\sqrt {-1}})}$ ${\displaystyle \mathbb {Q} (\mu _{3})=\mathbb {Q} (\mu _{6})=\mathbb {Q} ({\sqrt {-3}})}$

• A prime number is exactly then branched into if a divisor of is. is fully decomposed if and only if applies. ${\displaystyle p\neq 2}$ ${\displaystyle \mathbb {Q} (\mu _{n}),}$ ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle p\equiv 1{\pmod {n}}}$

• 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: ${\displaystyle n=\ell ^{\nu }}$ ${\displaystyle \zeta _{n}}$ ${\displaystyle n}$ ${\displaystyle \ell }$ ${\displaystyle \mathbb {Q} (\mu _{n})}$ ${\displaystyle \ell }$ ${\displaystyle 1-\zeta _{n}}$

${\displaystyle (\ell )=(1-\zeta _{n})^{\varphi (n)}.}$

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. ${\displaystyle \mathbb {Q} }$

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 ).