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

an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions...

In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical...

operations 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...

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in...

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator...

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

theorem) Algebraic function Closed-form expression Differential Galois theory Elementary function arithmetic Liouville's theorem (differential algebra) Tarski's...

Function field may refer to: Function field of an algebraic variety Function field (scheme theory) Algebraic function field Function field sieve Function...

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

In mathematics, an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection...

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

In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or...

of algebraic geometry codes are connected to algebraic function fields, the definitions of the codes are often given in the language of algebraic function...

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...

rational functions KX of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry...

occur in the algebraic function field case (not the number field case). Global L-functions can be associated to elliptic curves, number fields (in which...

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

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

of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties...

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called...

an inverse function, some facility was provided for algebraic manipulations of the natural logarithm even if it is not an algebraic function. The exponential...

n-dimensional projective algebraic variety over the field Fq with q elements and Nk is the number of points of V defined over the finite field extension Fqk of...

of an algebraic function field and representations of algebraic groups over the function field, generalizing class field theory of function fields from...

coefficients are algebraic numbers is again algebraic. That can be rephrased by saying that the field of algebraic numbers is algebraically closed. In fact...

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...

in algebraic geometry. For example, the dimension of an algebraic variety is the transcendence degree of its function field. Also, global function fields...

the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is a meromorphic function on the complex plane...

algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and...

function defined on M. More generally, given an algebraic variety V over some field K, the function field K(V), consisting of the rational functions defined...