Algebraiset

Algebraiset are mathematical objects characterized by algebraic relations, typically polynomial equations with coefficients in a base field. The term serves as an umbrella for several closely related notions, including algebraic numbers, algebraic sets, and algebraic functions, all of which are defined by polynomial constraints rather than by transcendental descriptions.

Algebraic numbers are complex numbers that satisfy a nonzero polynomial with integer coefficients. They form a countable set and include familiar numbers such as roots of unity and many radicals. The field consisting of all algebraic numbers is called the algebraic closure of the rationals.

Algebraic sets are subsets of affine or projective space defined as the common zeros of a family of polynomials. In algebraic geometry these sets are studied via their coordinate rings, morphisms, and geometric properties, with the Zariski topology providing a natural, coarse geometric framework.

Algebraic functions are functions that satisfy a polynomial equation with respect to the dependent variable. They need not be expressible by elementary formulas; examples include roots of polynomials and certain implicit functions arising from equations.

In computational contexts, algorithms for manipulating algebraiset include elimination theory, Gröbner bases, and resultant calculations. These tools allow solving polynomial systems, testing membership in algebraic sets, and performing symbolic reasoning in algebraic geometry and number theory. See also algebraic, algebraic geometry, and algebraic numbers for related concepts.