of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory...

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

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...

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,...

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

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...

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...

In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A...

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

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...

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

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...

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

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

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

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

mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus...

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...

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

Thus, every algebraic number field and the field of complex numbers C {\displaystyle \mathbb {C} } are of characteristic zero. The finite field GF(pn) has...

algebraic numbers. Consider the approximation of a complex number x by algebraic numbers of degree ≤ n and height ≤ H. Let α be an algebraic number of...

Universal algebra is still more abstract in that it is not interested in specific algebraic structures but investigates the characteristics of algebraic structures...

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...

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

In mathematics, the ideal class group (or class group) of an algebraic number field K is the quotient group JK /PK where JK is the group of fractional...

mathematics, if L is an extension field of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some...

algebraic groups, the spinor norm is a connecting homomorphism on cohomology. Writing μ2 for the algebraic group of square roots of 1 (over a field of...

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...