Test irreducibility of a polynomial
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irreducible(p) tests if the polynomial
A polynomial is irreducible over the field k if p is nonconstant and is not a product of two nonconstant polynomials in .
the polynomial is irreducible over the field implied by its coefficients.
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
or a univariate polynomial over an algebraic extension (see
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)))