Write down the element of f[a] a is the polynomial over x^3+2x+2 in gf(3)
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Use the Euclidean algorithm to find gcd( x8-1, x6-1) in Q[x] and write it as a linear combination of x8 -1 and x6 -1.
Solution: Let x8-1 = f(x) and x6-1 = g(x). We have f(x) = x2 g(x) + (x2-1), and g(x) = (x4+x2+1) (x2 -1), so this shows that gcd( x8-1, x6-1) = x2 - 1, and x2-1 = f(x) - x2 g(x).
(b) Factor x8-1 and x6-1 into products of polynomials that are irreducible over Q.
Solution: (Partial)
x8 - 1 = (x4-1)(x4+1) = (x-1)(x+1)(x2+1)(x4+1). The factor x2+1 is irreducible because it has no roots in Q; x4+1 is irreducible because substituting x+1 gives the polynomial x4+4x3+6x2+4x+2, which satisfies Eisenstein's criterion for p=2.
(c) Use your answer from (b) to compute gcd( x8-1, x6-1) in Q[x].
Solution: (x-1)(x+1)
Solution: Let x8-1 = f(x) and x6-1 = g(x). We have f(x) = x2 g(x) + (x2-1), and g(x) = (x4+x2+1) (x2 -1), so this shows that gcd( x8-1, x6-1) = x2 - 1, and x2-1 = f(x) - x2 g(x).
(b) Factor x8-1 and x6-1 into products of polynomials that are irreducible over Q.
Solution: (Partial)
x8 - 1 = (x4-1)(x4+1) = (x-1)(x+1)(x2+1)(x4+1). The factor x2+1 is irreducible because it has no roots in Q; x4+1 is irreducible because substituting x+1 gives the polynomial x4+4x3+6x2+4x+2, which satisfies Eisenstein's criterion for p=2.
(c) Use your answer from (b) to compute gcd( x8-1, x6-1) in Q[x].
Solution: (x-1)(x+1)
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