Test irreducibility of a polynomial

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irreducible(p) tests if the polynomial p is irreducible.

A polynomial is irreducible over the field k if p is nonconstant and is not a product of two nonconstant polynomials in .

irreducible returns TRUE if the polynomial is irreducible over the field implied by its coefficients. Otherwise, FALSE is returned. See the function factor for details on the coefficient field that is assumed implicitly.

The polynomial may be either a (multivariate) polynomial over the rationals, a (multivariate) polynomial over a field (such as the residue class ring IntMod(n) with a prime number n) or a univariate polynomial over an algebraic extension (see Dom::AlgebraicExtension).

Internally, a polynomial expression is converted to a polynomial of type DOM_POLY before irreducibility is tested.


Example 1

With the following call, we test if the polynomial expression x2 - 2 is irreducible. Implicitly, the coefficient field is assumed to consist of the rational numbers:

irreducible(x^2 - 2)

factor(x^2 - 2)

Since x2 - 2 factors over a field extension of the rationals containing the radical , the following irreducibility test is negative:

irreducible(sqrt(2)*(x^2 - 2))

factor(sqrt(2)*(x^2 - 2))

The following calls use polynomials of type DOM_POLY. The coefficient field is given explicitly by the polynomials:

irreducible(poly(6*x^3 + 4*x^2 + 2*x - 4, IntMod(13)))

factor(poly(6*x^3 + 4*x^2 + 2*x - 4, IntMod(13)))

irreducible(poly(3*x^2 + 5*x + 2, IntMod(13)))

factor(poly(3*x^2 + 5*x + 2, IntMod(13)))



A polynomial of type DOM_POLY or a polynomial expression

Return Values


Overloaded By