Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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 $f$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.