study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This...

x^{2}-x-1=0} , and the complex number 1 + i {\displaystyle 1+i} is algebraic as a root of X 4 + 4 {\displaystyle X^{4}+4} . Algebraic numbers include all integers...

an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More...

on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of...

generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of...

mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that is,...

In algebraic number theory, a quadratic field is an algebraic number field of degree two over Q {\displaystyle \mathbf {Q} } , the rational numbers. Every...

integers of an algebraic number field K {\displaystyle K} is the ring of all algebraic integers contained in K {\displaystyle K} . An algebraic integer is...

matroid. No good characterization of algebraic matroids is known, but certain matroids are known to be non-algebraic; the smallest is the Vámos matroid...

result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers...

real, although it is a field of real numbers. The totally real number fields play a significant special role in algebraic number theory. An abelian extension...

{\displaystyle K} form an algebraically closed field called an algebraic closure of K . {\displaystyle K.} Given two algebraic closures of K {\displaystyle...

{Q} } ⁠ are called algebraic number fields, and the algebraic closure of ⁠ Q {\displaystyle \mathbb {Q} } ⁠ is the field of algebraic numbers. In mathematical...

In number theory, an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by...

in the field of algebraic number theory, a modulus (plural moduli) (or cycle, or extended ideal) is a formal product of places of a global field (i.e....

fields: Algebraic number field: A finite extension of Q {\displaystyle \mathbb {Q} } Global function field: The function field of an irreducible algebraic curve...

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root...

\end{aligned}}} In general, this leads directly to the algebraic number field Q [ r ] {\textstyle \mathbb {Q} [r]} , which can be defined as the...

real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through...

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...

In mathematics, an algebraic function field (often abbreviated as function field) of n {\displaystyle n} variables over a field k {\displaystyle k} is...

algebraic number theory topics. These topics are basic to the field, either as prototypical examples, or as basic objects of study. Algebraic number field...

mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure...

if A is the ring of algebraic integers in an algebraic number field F (a finite extension of the rationals), then the algebraic K-groups of A are finitely...

abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions...

called self-adjoint. Archetypical examples of a *-ring are fields of complex numbers and algebraic numbers with complex conjugation as the involution. One...

elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined...

is an algebraic number. Fields of algebraic numbers are also called algebraic number fields, or shortly number fields. Algebraic number theory studies...

In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q {\displaystyle \mathbb {Q} } , the...

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which...