MathDB

For prime number

q

the polynomial

P(x)

with integer coefficients is said to be factorable if there exist non-constant polynomials

f_q,g_q

with integer coefficients such that all of the coefficients of the polynomial

Q(x)=P(x)-f_q(x)g_q(x)

are dividable by

q

; and we write:

P(x)\equiv f_q(x)g_q(x)\pmod{q}

For example the polynomials

2x^3+2,x^2+1,x^3+1

can be factored modulo

2,3,p

in the following way:

\left\{\begin{array}{lll} X^2+1\equiv (x+1)(-x+1)\pmod{2}\\ 2x^3+2\equiv (2x-1)^3\pmod{3}\\ X^3+1\equiv (x+1)(x^2-x+1) \end{array}\right.

Also the polynomial

x^2-2

is not factorable modulo

p=8k\pm 3

.
a) Find all prime numbers

p

such that the polynomial

P(x)

is factorable modulo

p

P(x)=x^4-2x^3+3x^2-2x-5

b) Does there exist irreducible polynomial

P(x)

\mathbb{Z}[x]

with integer coefficients such that for each prime number

p

, it is factorable modulo

p

Problem Statement