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I am trying to evaluate the polynomial (19v^2 + 49v + 8) raised to the 67th power where v^3 + 2 = 0. All computations are done mod 67.

Here's my first attempt:

R.<v> = PolynomialRing(GF(67))
poly = (19 * (v^2) + 49*v + 8)
after_frobenius = poly^67
after_frobenius(sqrt(-2))

But this gives the error: "positive characteristic not allowed in symbolic computation".

I suspect, I need to define the PolynomialRing over an extension field where v^3 + 2 = 0. But, I'm not sure how to do that. Any ideas?

I am trying to evaluate the polynomial (19v^2 + 49v + 8) raised to the 67th power where v^3 + 2 = 0. All computations are done mod 67.

Here's my first attempt:

R.<v> = PolynomialRing(GF(67))
poly = (19 * (v^2) + 49*v + 8)
after_frobenius = poly^67
after_frobenius(sqrt(-2))

But this gives the error: "positive characteristic not allowed in symbolic computation".

I suspect, I need to define the PolynomialRing over an extension field where v^3 + 2 = 0. But, I'm not sure how to do that. Any ideas?