In number theory, measure of non-unique factorization
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In number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK /PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of K. The order of the group, which is finite, is called the class number of K.
The theory extends to Dedekind domains and their fields of fractions, for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain.
divisor classgroup, or idealclassgroup, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can...
group. It is closely related to (though larger than) the idealclassgroup. The idele classgroup is not itself compact; the ideles must first be replaced...
Therefore, the idealclassgroup makes two fractional ideals equivalent if one is as close to being principal as the other is. The idealclassgroup is generally...
K and the Galois group of E over K is canonically isomorphic to the idealclassgroup of K using Frobenius elements for prime ideals in K. In this context...
extensions of K (in a fixed algebraic closure of K) and the generalized idealclassgroups defined via a modulus of K. It is called an existence theorem because...
generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely...
as the idealclassgroup. However, on a more general class of domains, including Noetherian domains and Krull domains, the idealclassgroup is constructed...
of integers O , {\displaystyle O,} group of fractional ideals J K , {\displaystyle J_{K},} and idealclassgroup Cl K = J K / K × . {\displaystyle \operatorname...
&d<0\\{\sqrt {|\Delta |}}/2&d>0.\end{cases}}} Then, the idealclassgroup is generated by the prime ideals whose norm is less than M K {\displaystyle M_{K}}...
abelian unramified extension of F, the Galois group of K over F is canonically isomorphic to the idealclassgroup of F. This statement was generalized to the...
Iwasawa theory is a deep relationship between p-adic L-functions and idealclassgroups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying...
theory), the size of the idealclassgroup of a number ring Class number (binary quadratic forms), the number of equivalence classes of binary quadratic forms...
\mathbb {Z} } has trivial idealclassgroup and unit group { ± 1 } {\displaystyle \{\pm 1\}} , thus each nonzero fractional ideal of Z {\displaystyle \mathbb...
infinite towers of number fields. It began as a Galois module theory of idealclassgroups, initiated by Kenkichi Iwasawa (1959) (岩澤 健吉), as part of the theory...
the Kronecker–Weber theorem. 1897 Weber introduces ray classgroups and general idealclassgroups. 1897 Hilbert publishes his Zahlbericht. 1897 Hilbert...
factorization is measured by the class number, commonly denoted h, the cardinality of the so-called idealclassgroup. This group is always finite. The ring...
principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the classgroup of an algebraic...
(Gras 1977) relates the p-parts of the Galois eigenspaces of an idealclassgroup to the group of global units modulo cyclotomic units. It was proved by Mazur...
Cyclotomic field Cubic field Biquadratic field Quadratic reciprocity Idealclassgroup Dirichlet's unit theorem Discriminant of an algebraic number field...
isomorphism classes of invertible sheaves on X themselves form an abelian group under tensor product. This group generalises the idealclassgroup. In general...
considerable class of prime exponents (see regular prime, idealclassgroup). His methods were closer, perhaps, to p-adic ones than to ideal theory as understood...
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