Test irreducibility of a polynomial
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irreducible(p
)
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.
With the following call, we test if the polynomial expression x^{2}  2 is irreducible. Implicitly, the coefficient field is assumed to consist of the rational numbers:
irreducible(x^2  2)
factor(x^2  2)
Since x^{2}  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 
p
content
 factor
 gcd
 icontent
 ifactor
 igcd
 ilcm
 isprime
 lcm
 poly
 polylib::divisors
 polylib::primpart
 polylib::sqrfree