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Considering a finite field function field $\mathbb{F}_p[x]$, we have $x^p = x$. If this were to be true, we would have finitely many unique polynomials in any finite field -- $p^{p}$ to be exact. For example, in $\mathbb{F}_2$ we would have $\{0,1,x,x+1\}$ as unique irreducibles as say $x^2+x+1 \equiv 1$. In this case, $Gal(\overline{\mathbb{F}_2}/ \mathbb{F}_2)$ would be trivial, but I assume this is not the case.

Is this correct? If it is not, why is that the case?

Jafari MM
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    Polynomials and polynomial functions are not the same thing – lhf Feb 25 '25 at 09:49
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    Yes. Over finite fields the number of monic irreducible polynomials is finite. But what you wrote, $;\Bbb F_p[x];$ is not a finite function field. It is the infinite ring of polynomials over $;\Bbb F_p;$ . The functions field is usually denoted by $;\Bbb F_p(x);$ – DonAntonio Feb 25 '25 at 09:51
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    It’s not that $x^p=x$ as a polynomial, but as values. Note that for elements in extensions of $\mathbb{F}_p$, $x^p\not= x$. – Michael Burr Feb 25 '25 at 09:51
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    For the "same" statement from the number theoretic study of the integers modulo primes, see Fermat's little theorem -- this is a statement about (coincident) values of powers of elements, not about a polynomial. – Eric Towers Feb 25 '25 at 09:54
  • Upvoting all the comments. If any of you can think of other threads that need to be linked to this one (in the dupe closure box), please @-ping me. I may be the only user with fitting dupehammer privileges, so I can tend to the matter :-) – Jyrki Lahtonen Feb 26 '25 at 05:05
  • Jafari, I think the first listed question addresses the main source of your confusion. You will find that you are not the only student with this problem. In fact, most of us needed a stop right at this point. Simply because our school level experience with polynomials did not need to make the distinction between polynomials and polynomial functions. For exactly the reason that over an infinite field two polynomials yielding the same function are necessarily the same polynomial - a claim that no longer holds over a finite field. – Jyrki Lahtonen Feb 26 '25 at 05:12

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