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Let $K=F_2[x]/(x^3+x+1)$. I want to show that $f(x)=x^4+x^2+1$ is reducible over $K$ but has no roots in it. How to proceed? I know that $F$ contains 8 elements, how is the structure of these elements? How can I realize the elements of this field?

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Excuse me, what is $F$? Do you mean $K$? – awllower Feb 3 '13 at 18:10
Note that $f(x)=x^4+x^2+1$ is reducible over $F_2$, it is a square. – Andreas Caranti Feb 3 '13 at 18:11
That's what I have to show, to split it as two factors of degree 2. – Mathematician Feb 3 '13 at 18:13
@WaqasAliAzhar, you should know the property of the map $a \mapsto a^2$ in characteristic 2. – Andreas Caranti Feb 3 '13 at 18:21
Great, @WaqasAliAzhar, you are nearly there. Now note that $x^2+x+1$ is irreducible in $F_2[x]$, and so is $x^3+x+1$, so that $\lvert K : F_2 \rvert = 3$. – Andreas Caranti Feb 3 '13 at 18:29

Hint: Find all quadratic polynomials over $\mathbb{F}_{2}$ and find two of which that when multiplied, and using $x^{3}+x+1\equiv0$, gives you $f$.

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