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...

the complex number 1 + i {\displaystyle 1+i} is algebraic because it is a root of x4 + 4. All integers and rational numbers are algebraic, as are all...

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

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...

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...

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,...

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...

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

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

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...

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

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

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...

mathematics, an algebraic function field (often abbreviated as function field) of n variables over a field k is a finitely generated field extension K/k...

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

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...

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...

commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are...

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...

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

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

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

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....

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...

an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are...

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...

objects (also called quotient algebras in universal algebra, and cokernels in category theory). For many types of algebraic structure, the fundamental theorem...

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 mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions...