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It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as irreducible over an algebraic closure. In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space.
Casus irreducibilis (from Latin 'the irreducible case') is the name given by mathematicians of the 16th century to cubic equations that cannot be solved in terms of real radicals, that is to those equations such that the computation of the solutions cannot be reduced to the computation of square and cube roots.
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials.The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong.
In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of ...
Here C r is the function space whose members are continuous functions with r continuous derivatives from a manifold M to a manifold N. The space C r (M, N), of C r mappings between M and N, is a Baire space, hence any residual set is dense. This property of the function space is what makes generic properties typical.
If f(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x 2 − 2 is irreducible over the rational numbers and has 2 {\displaystyle {\sqrt {2}}} as a root; hence there is no linear or constant polynomial over the rationals having 2 {\displaystyle {\sqrt {2}}} as a root.
In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field. [1] [2] [3] For example, + is absolutely irreducible, but while + is irreducible over the integers and the reals, it is reducible over the complex numbers as + = (+) (), and thus not absolutely irreducible.
Here are some examples. Every polynomial is associated to a unique monic polynomial. In particular, the unique factorization property of polynomials can be stated as: Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials.