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Let $p$ be prime. Let $p_1(x)=x^p-x$, $p_2(x)=0$ be two polynomials in $F_p[x]$. We know that $p_1=p_2$ as functions, by Fermat's little theorem. However, can I say that $p_1(x)$ is the zero polynomial?

Tongou Yang
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    Nope. $p_1$ is a nonzero polynomial of degree $p$ with the leading coefficient $1$. –  Jan 12 '18 at 22:37
  • See here for more discussion about the difference between polynomial functions and (formal) polynomials. In calculus and below students really only encounter polynomial functions. Which is ok because over infinite fields there is no difference. It may be helpful to think that into a polynomial from $\Bbb{F}_p[x]$ you can plug-in as $x$, in addition to elements of $\Bbb{F}_p$ also elements of any extension field (elements of any extension ring actually, such as another polynomial, or a matrix with entries in $\Bbb{F}_p$). – Jyrki Lahtonen Jan 14 '18 at 10:48

2 Answers2

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No. The elements of a polynomial ring like $\mathbb{F}_p[x]$ are polynomials as formal expressions, not as functions. In other words, two polynomials are defined to be the same only when all of their coefficients are the same. (Or if you prefer, a polynomial is defined to be its coefficients: an element of $\mathbb{F}_p[x]$ is technically a function $f:\mathbb{N}\to \mathbb{F}_p$ such that $f(n)=0$ for all but finitely many $n\in\mathbb{N}$. We normally write such a function as the "polynomial expression" $f(0)+f(1)x+f(2)x^2+f(3)x^3+\dots$.)

Eric Wofsey
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No. They give the same functions $\Bbb Z_p\to\Bbb Z_p$ (the zero function) but they are not the same polynomials.

Arthur
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  • Which can be verified by taking the derivative. – Gyro Gearloose Jan 12 '18 at 22:39
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    @Gyro You wouldn't need to go that far. Simply the definition of polynomial as a finite linear combination of the formal symbols $x^i$. Different linear combinations are different polynomials, period. – Arthur Jan 12 '18 at 22:43
  • Yes, but they are not only unique by definition, they work out differently, and I think this is worth to mention. – Gyro Gearloose Jan 12 '18 at 22:49
  • Of course, if the two are different by definition, there are well-defined functions with the set of polynomials as domain that take them to different things. Differentiation is such a function with the added bonus that it is useful. – Arthur Jan 12 '18 at 22:54