Cohn's irreducibility criterion
From Number
Statement
Suppose is a polynomial with integer coefficients, i.e.,
. Suppose that all the coefficients of
are nonnegative. Further, suppose
is a natural number strictly greater than all coefficients. Then, if
is a prime number,
must be an irreducible polynomial.
An alternate formulation is as follows: for any , if a number with digits
written in base
is prime (so in particular
for
) then the polynomial
is irreducible.
Related facts
- Bunyakovsky conjecture is a conjectured converse of sorts: if a polynomial is irreducible and the set of its values does not have a gcd, then the polynomial must take prime values at infinitely many natural numbers.