Let F_(256) be the finite field with 2^8=256 elements. Consider the polynomial over this field x^2+x+1

garkochenvz

garkochenvz

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2022-08-18

Let F 256 be the finite field with 2 8 = 256 elements. Consider the polynomial over this field
x 2 + x + 1.

Answer & Explanation

nedervdq3

nedervdq3

Beginner2022-08-19Added 13 answers

The method you cite should work, because the polynomial is of degree 2 and hence any factors will correspond to solutions of the equation x 2 + x + 1 = 0. But you have to do it right and work over the field F 256 , which means you must begin by constructing that field etc. Working over the integers modulo 256 will not do.
Here is my suggestion to avoid this work: the polynomial is irreducible over F 2 but it becomes reducible over F 4 and extensions thereof. Since 256 = 2 8 and 8 is even, it contains F 4 as a subfield and hence the polynomial is reducible.

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