How to find the root of the polynomial $x^2+x+1$ over $\mathbb{Z}_2$ in this field?

Question

I am having some troubles understanding the proof for a statement. The question is:

Suppose R is the polynomial ring $\mathbb{Z}_2[x]$. Let $(x^4+x+1)$, I, be the principal ideal of this ring. Therefore, the quotient ring $\frac{\mathbb{Z}_2[x]}{I}$, F, is a field as $x^4+x+1$ is irreducible.

Then, the question asks to prove that:

every quadratic polynomial over $\mathbb{Z}_2[x]$ has a root in F.

It is easy to prove for reducible quadratic polynomial because they have roots $I+0$ or $I+1$. So, what we only need to consider is the polynomial $x^2+x+1$, which is irreducible over $\mathbb{Z}_2[x]$.

The proof directly says that $I+x^5 \in F$ is such a root of $x^2+x+1$. It is not clear for me why $I+x^{10}+x^5+1 = I$ in this case. This is confusing for me. Please give me some help on this part, thanks!

Plus, the proof before states that ''if $\alpha \in F$, then $\alpha^{15}=1$'', which I understand but don't know how to use.

The element $x+I$ is not a root of $x^2+x+1=0$. Try $x^5+I$. — Angina Seng, Dec 25 '18 at 17:32
I think $F$ only contains cosets $I+p(x)$ where the degree of p(x) is less than 4. — Jamie Carr, Dec 25 '18 at 17:39
Jamie, every coset of $I$ contains a unique representative of degree less than 4. Find the one in the coset of $x^5$ and you will be done. — Jyrki Lahtonen, Dec 25 '18 at 17:42
And, if you get stuck, you can take a look at this Q&A I prepared for referrals. You see, the zeros of $x^2+x+1$ are in the subfield of four elements (do you see why?), and I had to calculate those in the last example. — Jyrki Lahtonen, Dec 25 '18 at 17:44
@LordSharktheUnknown Thank you, if I'm not misunderstanding your point, I think $I+x^5=I+x^2+x$ in this case? Plus, how did you come up with x^5 in advance? — Jamie Carr, Dec 25 '18 at 17:46

score 1 · Answer 1 · answered Dec 25 '18 at 19:54

In the following GF(q) indicates the unique finite field or order q.

Use degrees and the uniqueness of finite fields of a given order. Since $x^4+x+1$ is irreducible, $GF(2)[x]/I$ has degree 4 over $GF(2)$ and thus has order $2^4=16$, and so is $GF(16)$. Irreducible quadratic polynomials over $GF(2)$ split in a degree 2 extension, which has order $2^2=4$ and thus is $GF(4)$. By uniqueness of Galois fields, any degree 2 extension of $GF(4)$ must be $GF(16)$ since $4^2=16$. Thus $GF(4) \subset GF(16)=F$ and any quadratic over $GF(2)$ splits over $GF(4)$, so we are done.

$a^{15}=1$ for all $a\in F$ since the multiplicative group of $F$ has order $16-1=15$

How to find the root of the polynomial $x^2+x+1$ over $\mathbb{Z}_2$ in this field?

1 Answers1