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Galois Theory — Thomas Judson
Mathematics — Abstract Algebra · finite-fields-factoring-polynomials
Problem Factor each of the following polynomials in Z 2 [ x ] \mathbb{Z}_2[x] Z 2 [ x ] x 5 − 1 x^5 - 1 x 5 − 1 x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 x 9 − 1 x^9 - 1 x 9 − 1 x 4 + x 3 + x 2 + x + 1 x^4 + x^3 + x^2 + x + 1 x 4 + x 3 + x 2 + x + 1 x 5 − 1 = ( x + 1 ) ( x 4 + x 3 + x 2 + x + 1 ) x^5 - 1 = (x+1)(x^4 + x^3 + x^2 + x + 1) x 5 − 1 = ( x + 1 ) ( x 4 + x 3 + x 2 + x + 1 ) x 9 − 1 = ( x + 1 ) ( x 2 + x + 1 ) ( x 6 + x 3 + 1 ) x^9 - 1 = (x+1)(x^2 + x + 1)(x^6 + x^3 + 1) x 9 − 1 = ( x + 1 ) ( x 2 + x + 1 ) ( x 6 + x 3 + 1 ) Browse the problem library