I was looking at this post and saw a comment left without an answer that intrigued me. The question posed was about a field being isomorphic to a quotient ring and an answer showed that: $$\frac {\mathbb Z[x,y]}{(5,x^2-y,xy+x+1)}\cong \frac {\mathbb Z_5[x,y]}{(x^2-y,xy+x+1)}\cong\frac {\mathbb Z_5[x]}{(x^3+x+1)} $$ The comment left asked if there was an easy way to see if the polynomial $x^3+3x+2$ has a root in a finite field of $125$ elements. Is there a general way to show this for any function and any finite field?
Knowing is a polynomial has a root in a finite field of $125$ elements
abstract-algebrafield-theoryfinite-fields
Best Answer
$x^3+3x+2$ has no root mod $5$ and so is irreducible. Therefore, $\frac {\mathbb Z_5[x]}{(x^3+x+1)}$ is a field with $5^3=125$ elements which contains a root of $x^3+3x+2$.
Things are not always so easy. In general, a polynomial $f$ has a root in a finite field of order $q=p^n$ iff $\gcd(f,x^q-x)\ne1$ mod $p$. This can be settled with Euclid's algorithm, but is not suitable for hand computation for large $q$ and $f$.