unramified extension

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• 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

• Herbrand–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… …   Wikipedia

• Abhyankar's lemma — In mathematics, Abhyankar s lemma (named after Shreeram Shankar Abhyankar) allows one to kill tame ramification by taking an extension of a base field. More precisely, Abhyankar s lemma states that if A , B , C are local fields such that A and B… …   Wikipedia

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• Henselian ring — In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel s lemma holds. They were defined by harvtxt|Azumaya|1951, who named them after Kurt Hensel. Some standard references for Hensel rings are… …   Wikipedia

• Igor Shafarevich — Igor Rostislavovich Shafarevich (Russian: Игорь Ростиславович Шафаревич, born June 3, 1923 in Zhytomyr) is a Russian mathematician, founder of the major school of algebraic number theory and algebraic geometry in the USSR, and a political writer …   Wikipedia

• Witt vector — In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order p is the …   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