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In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic...

See also irreducible component, algebraic variety. In universal algebra, irreducible can refer to the inability to represent an algebraic structure as...

two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions...

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

polynomial of a plane algebraic curve is irreducible, then one has an irreducible plane algebraic curve. Otherwise, the algebraic curve is the union of...

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...

is supposed to be irreducible. When this is not the case, the hypersurface is not an algebraic variety, but only an algebraic set. It may depend on the...

\operatorname {Spec} (k)} . An irreducible algebraic set is called an algebraic variety. algebraic space An algebraic space is a quotient of a scheme...

representation theory of groups and algebras, an irreducible representation ( ρ , V ) {\displaystyle (\rho ,V)} or irrep of an algebraic structure A {\displaystyle...

An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers...

linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over...

In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the...

affine algebraic set defined by equations with coefficients in a field K is absolutely irreducible if it is not the union of two algebraic sets defined...

{\displaystyle k(Y)} ). We see that the degree of an algebraic function field is not a well-defined notion. The algebraic function fields over k form a category; the...

structures all admit order-theoretic as well as algebraic descriptions. The sub-field of abstract algebra that studies lattices is called lattice theory...

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

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

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

from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a...

Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied...