Question: How to find roots of polynomial in finite field and extension finite field?

Hi EveryOne!

I have polynomial: p(x) = x^4 + 27x^3 + x^2 + 16x +1 over finite field F=GF(2^8)/f(x)=x^8 + x^4 +x^3 +1

The factors of this polynomial are: (x + 37)(x + 217)(x^2 + 213x +30) (in maple)

Hence there two roots of p(x): x = 37 and x = 217 in GF(2^8). The factor x^2 + 231x +30 is of degree 2. There are not roots in F. But in extension field GF((2^8)^2) of F, also there are two roots of factor x^2 + 213x + 30 (for example: x = 256 and x = 256^256 = 487).

How to find these roots of p(x) in extension field GF((2^8)^2) by maple? Please help me! Thanks a lot.

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