As in the title, I'm trying to understand the implications of Fermat's Little Theorem in $\Bbb Z/p\Bbb Z [x] $. Fermat states that if $p $ is prime then for all integer $x$, $x^p-x $ is a multiple of $p$. As far as I can see, this should mean that $x^p-x=0$ in $\Bbb Z/p\Bbb Z [x] $, but I'm not sure. Is that correct?

  • $\begingroup$ No, it does not hold for the variable in the polynomial ring, since the statement is for an integer. $\endgroup$ – Tobias Kildetoft Jul 18 '17 at 19:19
  • $\begingroup$ Polynomials and the functions they determine by evaluation should not be confused, especially over finite rings. The former family is infinite, the latter finite. $\endgroup$ – Ravi Jul 18 '17 at 19:20
  • $\begingroup$ @DougM Yeah, I removed my comment and made a new one when I saw you had removed it :) $\endgroup$ – Tobias Kildetoft Jul 18 '17 at 19:21
  • $\begingroup$ @Tobias I see, thank you $\endgroup$ – Richard Jul 18 '17 at 19:22
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    $\begingroup$ See also this answer for some background on formal polynomials vs. polynomial functions. $\endgroup$ – Bill Dubuque Jul 18 '17 at 21:43

The polynomial $x^p-x$ is not the zero polynomial in $\Bbb Z/p\Bbb Z [x]$.

The polynomial function $x^p-x$ is the zero function $\Bbb Z/p\Bbb Z \to \Bbb Z/p\Bbb Z$. That's what Fermat says.

Polynomials and polynomial functions are different objects but they can be identified when the base field is infinite.


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