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irreducible -- test irreducibility of a polynomial

Introduction

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

Call(s)

irreducible(p)

Parameters

p - a polynomial of type DOM_POLY or a polynomial expression

Returns

TRUE or FALSE.

Overloadable:

p

Related Functions

content, factor, gcd, icontent, ifactor, igcd, ilcm, isprime, lcm, poly, polylib::divisors, polylib::primpart, polylib::sqrfree

Details

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)
                                   TRUE
>> factor(x^2 - 2)
                                   2
                                  x  - 2

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

>> irreducible(sqrt(2)*(x^2 - 2))
                                   FALSE
>> factor(sqrt(2)*(x^2 - 2))
                         1/2       1/2        1/2
                        2    (x + 2   ) (x - 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)))
                                   TRUE
>> factor(poly(6*x^3 + 4*x^2 + 2*x - 4, IntMod(13)))
                       3      2
               6 poly(x  + 5 x  - 4 x - 5, [x], IntMod(13))
>> irreducible(poly(3*x^2 + 5*x + 2, IntMod(13)))
                                   FALSE
>> factor(poly(3*x^2 + 5*x + 2, IntMod(13)))
        3 poly(x + 5, [x], IntMod(13)) poly(x + 1, [x], IntMod(13))

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