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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 abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial. In representation theory, an irreducible representation is a nontrivial representation with no nontrivial proper subrepresentations. Similarly, an irreducible module is another name for a simple module.
If is an irreducible polynomial of prime degree with rational coefficients and exactly two non-real roots, then the Galois group of is the full symmetric group. [2] For example, f ( x ) = x 5 − 4 x + 2 ∈ Q [ x ] {\displaystyle f(x)=x^{5}-4x+2\in \mathbb {Q} [x]} is irreducible from Eisenstein's criterion.
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.
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible.
The fact that the polynomial after substitution is irreducible then allows concluding that the original polynomial is as well. This procedure is known as applying a shift . For example consider H = x 2 + x + 2 , in which the coefficient 1 of x is not divisible by any prime, Eisenstein's criterion does not apply to H .
Cohn's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in [] —that is, for it to be unfactorable into the product of lower-degree polynomials with integer coefficients.
In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of and is not a divisor of for any k < n. Its roots are all n th primitive roots of unity e 2 i π k n {\displaystyle e^{2i\pi {\frac {k}{n}}}} , where k runs over the positive integers less ...