Re: Algebra with root and Z_2[x].
- From: José Carlos Santos <jcsantos@xxxxxxxx>
- Date: Fri, 23 Nov 2007 07:54:21 +0000
On 23-11-2007 7:06, 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>.
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x^4 + x + 1 is irreducible over Z_2.
By Kronecker, x^4 + x + 1 has a root in Z_2[x] / <x^4 + x + 1>.
Anyway, this is not useful in my problem.
Let f(x) = x + a in Z_2[x].
a is 0 or 1.
so, f(x) has a root in Z_2.
Let f(x) = x^2 + a.x + b in Z_2[x].
Sorry. I can't progress any more.
so, I need your advice.
You have four possibilities: f(x) = x^2, f(x) = x^2 + x, f(x) = x^2 + 1,
and f(x) = x^2 + x + 1. The first three have a root in Z_2 already. Now,
take a = [x] in Z_2[x]/<x^4 + x + 1>. Then
a^2 + a + 1 = [x^2 + x + 1]
and therefore
(a^2 + a + 1)^2 = [x^4 + x^2 + 1] = [x^2 + x] = (a^2 + a + 1) + 1.
So, a^2 + a + 1 is a root of x^2 + x + 1 in Z_2[x]/<x^4 + x + 1>.
Now, try the same approach with fourth degree polynomials.
Best regards,
Jose Carlos Santos
.
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