algebraiczne

Algebraiczne is a Polish adjective meaning algebraic. It is used in mathematics to describe objects or relations that are defined by algebraic rules, typically involving polynomials with coefficients in a given field, usually the rational numbers. In Polish mathematical vocabulary, it appears in phrases such as liczby algebraiczne (algebraic numbers), równania algebraiczne (algebraic equations), and geometria algebraiczna (algebraic geometry).

Liczby algebraiczne are numbers that are roots of a nonzero polynomial with rational coefficients. This includes all integers and rational numbers, and also many irrational numbers such as the square root of 2 or the cube root of 5. The set of algebraic numbers is countable and forms a field. The field of all algebraic numbers is algebraically closed, meaning every polynomial with coefficients in this field has a root also in the field. By contrast, most real or complex numbers are transcendental, not roots of any such polynomial.

In broader mathematical usage, algebraiczne describes elements in areas like algebraic equations, algebraic number theory, and algebraic geometry. In algebraic geometry, for example, one studies solution sets of polynomial equations, called algebraic varieties, and their geometric properties.

Etymology traces algebra to medieval and later Latin and Arabic roots; in Polish, the related adjective form is algebraiczny, with algebraiczne used to agree with plural or neuter nouns such as liczby algebraiczne. The term denotes a standard, widely used class of objects defined by polynomial relationships.