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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.
then Q is irreducible over the rational numbers. It will also be irreducible over the integers, unless all its coefficients have a nontrivial factor in common (in which case Q as integer polynomial will have some prime number, necessarily distinct from p, as an irreducible factor).
Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. 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 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 irreducible two-dimensional representation of the symmetric group S 3 of order 6, originally defined over the field of rational numbers, is absolutely irreducible. The representation of the circle group by rotations in the plane is irreducible (over the field of real numbers), but is not absolutely irreducible. After extending the field to ...
The two-dimensional irreducible complex representation described above gives the quaternion group Q 8 as a subgroup of the general linear group (,). The quaternion group is a multiplicative subgroup of the quaternion algebra:
A Hurwitz integer is irreducible if and only if its norm is a prime number. The irreducible quaternions are sometimes called prime quaternions, but this can be misleading as they are not primes in the usual sense of commutative algebra: it is possible for an irreducible quaternion to divide a product ab without dividing either a or b. Every ...
Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable (which is the case for every p but a finite number) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P.