Loop over integers mod p [closed]

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It would be interesting to know what code you tried and what error you got.

This would allow to explain precisely what failed in what you tried and why.

The following works for me.

sage: for p in primes_first_n(100):
....:     x = polygen(GF(p))
....:     print("%4d %s" %(p, (x^2 - 17).factor()))
....:

This runs through the first 100 primes, and for each of them declares x as the generator for a polynomial ring over the finite field with p elements, and prints out the factorization of the polynomial x^2 - 17 in that ring.

Note that Zmod(p) or Integers(p) creates the ring of integers modulo p, considered with its ring structure, while GF(p) (for Galois Field) creates it as a field.