unramified ideal
1Ideal class group — In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field (or more generally any Dedekind domain) can be described by a certain group known as an ideal class group (or class group). If… …
2Principal ideal — In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R .More specifically: * a left principal ideal of R is a subset of R of the form R a := { r a : r in R }; * a… …
3Glossary of scheme theory — This is a glossary of scheme theory. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of… …
4Discriminant of an algebraic number field — A fundamental domain of the ring of integers of the field K obtained from Q by adjoining a root of x3 − x2 − 2x + 1. This fundamental domain sits inside K ⊗QR. The discriminant of K is 49 = 72.… …
5Hilbert class field — In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K . Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal… …
6Frobenius endomorphism — In commutative algebra and field theory, which are branches of mathematics, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of rings with prime characteristic p , a class importantly including fields. The… …
7Algebraic number field — In mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector… …
8Splitting of prime ideals in Galois extensions — In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of… …
9Chebotarev's density theorem — in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field Q of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic… …
10Herbrand–Ribet theorem — In mathematics, the Herbrand–Ribet theorem is a result on the class number of certain number fields. It is a strengthening of Kummer s theorem to the effect that the prime p divides the class number of the cyclotomic field of p th roots of unity… …
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