Re: Algebra with root and Z_2[x].

On Fri, 23 Nov 2007, mina_world wrote:

Prove that every polynomial of degree 1, 2, or 4 in Z_2[x]
has a root in Z_2[x] / <x^4 + x + 1>.

If P(x) has degree 1, then P(x) = x + a, which has root.
If P(x) has degree 2, then P(x) = x^2 + ax + b.
Case x^2 + ax + 1
subcase x^2 + 1 = (x + 1)^2, has root.
subcase x^2 + x + 1, has root x^2 + x + 1

x^4 + x^2 + 1 + x^2 + x + 1 + 1

If P(x) has degree 4, then P(x) = x^4 + ax^3 + bx^2 + cx + d.
Case x^4 + ax^3 + bx^2 + cx + 1
subcase a = c = 0. Use previous results and
(x^2 + bx + 1)^2 = x^4 + bx^2 + 1

????
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