Distinct zeros of polynomials in $5$-adics.

Question

How many distinct zeros does each of the following polynomials have in $\mathbb{Z}_5$?

1. $f(x) = x^3 + 5x + 5$;
2. $g(x) = x^5 + 2$;
3. $h(x, y) = x^2 + y^2$.

I know how to do the first two, but I'm stuck on the third one; it seems that the same tricks do not work. Can someone give me a hint?

Answer 1

As you said, we need to use a completely different trick from 1 and 2. If we think of the equation in $\mathbb{C}$, then we know that$$x^2 + y^2 = (x + \sqrt{-1}y)(x-\sqrt{-1}y).$$Next, as we know that $\sqrt{-1}$ is in $\mathbb{Z}_5$ (use Hensel's lemma with $2^2 = -1\text{ (mod }5\text{)}$), then we know that in $\mathbb{Z}_5$ this factorization also exists. Hence, given any nonzero $x$ in $\mathbb{Z}_5$, I can give you two different $y$ in $\mathbb{Z}_5$ such that $x^2 + y^2 = 0$, while if $x = 0$, then we have $y= 0$.

Answer 2

By Hensel Lemma, there exists some $i \in \Bbb{Z}_5$ such that $i^2+1=0$.

Factorize $$x^2+y^2 = (x+iy)(x-iy)$$ to realize that all pairs $\{ (a, ia) : a \in \Bbb{Z}_5 \}$ are roots of $x^2+y^2$ (so they are infinitely many).