In mathematics, a ring class field is the abelian extension of an algebraic number field K associated by class field theory to the ring class group of some order O of the ring of integers of K.[1]
^Frey, Gerhard; Lange, Tanja (2006), "Varieties over special fields", Handbook of elliptic and hyperelliptic curve cryptography, Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, Florida, pp. 87–113, MR 2162721. See in particular p. 99.
mathematics, a ringclassfield is the abelian extension of an algebraic number field K associated by classfield theory to the ringclass group of some...
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue classring, is a construction quite...
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...
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from...
F} is an injective ring homomorphism from R {\displaystyle R} into a field F {\displaystyle F} , then there exists a unique ring homomorphism g : Frac...
mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of classfield theory, a branch of...
of Discriminant of an algebraic number field § Definition. For real quadratic integer rings, the ideal class number, which measures the failure of unique...
integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division...
ring of integers of K is a unique factorization domain, in particular if K = Q {\displaystyle K=\mathbb {Q} } , then K is its own Hilbert classfield...
In mathematics, classfield theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions...
factorization domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains...
principal ring or, equivalently, if K {\displaystyle K} has class number 1. Given a number field, the class number is often difficult to compute. The class number...
ring homomorphisms S → T and R → S is a ring homomorphism R → T. For each ring R, the identity map R → R is a ring homomorphism. Therefore, the class...
categories in mathematics, the category of rings is large, meaning that the class of all rings is proper. The category Ring is a concrete category meaning that...
the definition: see below. A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however...
of Q with trivial class group, the ring of integers is Euclidean (not necessarily with respect to the absolute value of the field norm; see below). Assuming...
the case when R is local, making local rings a particularly deeply studied class of rings. The residue field of R is defined as k = R / m. Any R-module...
integral domains are given with the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains...
as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization...
ring is an integral domain D such that for every non-zero element x of its field of fractions F, at least one of x or x−1 belongs to D. Given a field...
a field extension as an injective ring homomorphism between two fields. Every non-zero ring homomorphism between fields is injective because fields do...
all fields are principal ideal domains. Principal ideal domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains...
theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional...
number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their...
well. A ring whose localizations at all prime ideals are integrally closed domains is a normal ring. Let A be an integrally closed domain with field of fractions...
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