Dear all,
I understand this exercise as follows: we let f = Phi_{13} be the cyclotomic polynomial x^{12} + ... + x + 1 and reduce f modulo p to get a polynomial g with coefficients in F_p. Then, we are to show that g is a reducible polynomial.
My problem is that I don't think this is true. Some counterexamples I think I found using a computer algebra software are p = 2, 7, 11, 41, 59, ... Essentially, I think g is reducible if and only if p is not a generator of the multiplicative group (Z / 13 Z)*, but there are plenty of primes that do generate this group as shown above.
Maybe someone can back me up or correct a mistake in my thinking. Many thanks!
Best,
Julian