I have a 6th degree polynomial with unknown coefficients, but I need to take it (mod x^4+1) to find some 3rd degree polynomial. Using wolfram alpha and mathematica 9.0, I could only reduce the polynomial to a different 6th degree polynomial that was shorter.

e.g. is what I'm given, where each a_i is something like:

...

for some unknown .

Also, the coefficients are to (mod 2).

How can I find (mod x^4+1) as a 3rd degree polynomial?

I heard that under mod (x^4 + 1) and (mod 2), I have congruence x^4 = -1 = 1 but that doesn't make sense to me?