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Over a finite field, such as $\mathbb{Z}/p\mathbb{Z}$, the number of roots of a polynomial is no larger than the degree. I'm interested in how does this generalize to $\mathbb{Z}/p^k\mathbb{Z}$.

I'm sure that this has been looked at, but I haven't been able to locate any relevant literature. Thanks!

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    $\begingroup$ The polynomial $x^2$ has precisely $p$ roots when $k= 2$. So any answer will involve $p$, not just the degree of the polynomial. If the reduction of the polynomial is separable, then you can use Hensel's Lemma to find the number of roots. $\endgroup$
    – Jason Starr
    Commented Mar 22, 2018 at 18:51
  • $\begingroup$ Sure, that's a good point. But what I'm interested in is some theory, or at the very least a structural result, a theorem, you know, something beyond mere doodles :) $\endgroup$
    – user122270
    Commented Mar 22, 2018 at 18:57
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    $\begingroup$ I suggest you pick up any book with the statement of Hensel's Lemma and read that statement. $\endgroup$
    – Jason Starr
    Commented Mar 22, 2018 at 18:57
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    $\begingroup$ There's a classical theorem of Nagell and Ore in this direction. For an improved modern version, see Corollary 2 of Stewart's paper "On the number of solutions of polynomial congruences and Thue equations": uwaterloo.ca/pure-mathematics/sites/ca.pure-mathematics/files/… $\endgroup$
    – so-called friend Don
    Commented Mar 22, 2018 at 19:16
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    $\begingroup$ Maybe you look for the Igusa zeta function? $\endgroup$
    – Chris Wuthrich
    Commented Mar 22, 2018 at 23:49

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