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Reducibility over a certain field.

I am new to field theory. How can I show that $X^4+X^2+1$ has no roots in $F_2[X]/(X^3+X+1)$? All I know at this moment is that it is reducible over $F_2[X]/(X^3+X+1)$ as $X^4+X^2+1=(X^2+X+1)^2$. How to proceed with this problem?

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marked as duplicate by DonAntonio, Matthew Pressland, Davide Giraudo, rschwieb, Thomas Feb 4 '13 at 14:26

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up vote 2 down vote accepted

Since $X^4+X^2+1=(X^2+X+1)^2$, the roots of $X^4+X^2+1$ are also roots of $X^2+X+1$, which is irreducible over $\Bbb F_2$. This being a quadratic polynomial, it remains irreducible in the cubic extension $\Bbb F_2[X]/(X^3+X+1)$, all whose elements have minimal polynomials of degree $1$ or $3$ (a root of $X^2+X+1$ would have that polynomial as minimal polynomial and therefore span a subfield of degree $2$, but there aren't any such subfields.)

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Hint: The roots of $x^{4}+x^{2}+1$ are exactly the roots of $x^{2}+x+1$ and the latter is a polynomial of degree $2$.

What is $[\mathbb{F}_{2}[x]/\langle x^{3}+x+1\rangle:\mathbb{F}_{2}]$ ?

If $K$ is the splitting field of $x^{2}+x+1$, what is $[K:\mathbb{F}_{2}]$ ?

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