Let $K = \mathbb F_p[x]/f(x)$. Find all roots of $f(t)$ in $K$. ($p$ is prime and $f(x)$ is irreducible)

Question

This question confuses me. I'm not sure how to go about finding the roots of the polynomial in the field extension K. Just to clarify I have an example if it helps but I just need to understand. Let $K = \mathbb F_p[x]/(x^3 + 3x^2 + 3x + 3)$. Find all the roots of $f(t) = t^3 + 3t^2 + 3t + 3$ in the field $K$.

Thank you

Answer 1

In general, to find the roots of a polynomial $q(t) \in K[t]$, where $K = \mathbb{Z}_{p}[x] / (f(x))$ for some irreducible $f(x) \in \mathbb{Z}_{p}[x]$, $p$ a prime number, you have to observe that $K = \{ \sum_{i=0}^{\deg(f)-1} a_{i} x^{i} : a_{i} \in \mathbb{Z}_{p} \}$. Then you have to substitute for all $h(x) \in K$, and evaluate $q \circ h(x)$, and check if $p(x) \mid q \circ h(x)$, if so then $h(x)$ is a root. In your case, there are $7^{\deg(f)} = 343$ possibilities. Also, we know there is $3$ roots because $p(t)$ is degree 3 in your case, and $t = x$ is a root. If one does a fair amount of computation, one gets that $p(t) = (t-x)(t-(2x+1))(t-(4x+3)) \in K[t]$. So the roots are $x,2x+1,4x+3 \in K$.

Answer 2

Suppose that $p$ is a prime such that $x^3 +3x^2 +3x +3$ is irreducible over $\mathbb{F}_p$. Then we have that $K = \mathbb F_p[x]/(x^3 + 3x^2 + 3x + 3)= \mathbb F_p[\alpha]$, where $\alpha=x+ (x^3 + 3x^2 + 3x + 3)$.

Clearly one root of $t^3 +3 t^2 + 3 t + 3$ is $\alpha$.

Since for any member $c$ of $\mathbb{F}_p$ we have that $c^p=c$, we will get $$ 0=0^p=(\alpha^3 +3\alpha^2+3\alpha+3)^p= ((\alpha^p)^3+3(\alpha^p)^2+3(\alpha^p)+3), $$ so that $\alpha^p$ is another root, and then so is $\alpha^{p^2}$.

That is the roots are $\alpha,\alpha^p,\alpha^{p^2}$.

(If you wonder why we can stop here, note that since $K^*$ is cyclic of order $p^3-1$, we will get $\alpha^{p^3}=\alpha$.)

For many purposes that is a perfectly useful way to express the roots.

If you really need in some concrete case to express them as polynomials of degree less than $3$ in $\alpha$ all you need do is calculate the remainder when $X^p$ is divided by $X^3+3X^2+3X+3$ using your favourite algorithm.

All this would apply starting with any irreducible polynomial $f(x)$.