unramified field
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Algebraic 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… … Wikipedia
Discriminant 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.… … Wikipedia
Hilbert 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… … Wikipedia
Genus field — In algebraic number theory, the genus field G of a number field K is the maximal abelian extension of K which is obtained by composing an absolutely abelian field with K and which is unramified at all finite primes of K . The genus number of K is … Wikipedia
Conductor (class field theory) — In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map. Contents 1 Local… … Wikipedia
Glossary 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… … Wikipedia
Frobenius 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… … Wikipedia
Galois module — In mathematics, a Galois module is a G module where G is the Galois group of some extension of fields. The term Galois representation is frequently used when the G module is a vector space over a field or a free module over a ring, but can also… … Wikipedia
Chebotarev'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… … Wikipedia
Splitting 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… … Wikipedia
Étale morphism — In algebraic geometry, a field of mathematics, an étale morphism (pronunciation IPA|) is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem,… … Wikipedia
