# Tag Info

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### The maximal subset of a finite field where the sum of any subset is non-zero

I could trace this question down to the paper of Erdős and Heilbronn "On the addition of residue classes mod $p$" (Acta Arith. 17 (1970), 227–229), where it is shown that for a prime $p$, if ...

### Is $x^p-x+1$ always irreducible in $\mathbb F_p[x]$?

You might be interested in Artin-Schreier theory.
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### Finite field "contour" sum

What you call finite field contour sums are more usually called exponential sums. Like complex analytic contour integrals, they do not always (or even usually) have a nice exact formula. Instead, ...
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### Algebraic dynamics in finite fields

There is a growing body of literature on dynamics of rational maps over finite fields. The following paper would seem to be relevant. Ugolini, S., Graphs associated with the map $x\mapsto x+x^{-1}$ ...
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### Can you use Chevalley‒Warning to prove existence of a solution?

Let $f_i \in \mathbb{F}_q[X_1,\dots,X_n]$ have degree at most $d$ for each $i \in [|1,r|]$, and assume that the affine scheme  X = \operatorname{Spec} \mathbb F_q [x_1,\ldots,x_n]/(f_1, \ldots, f_r) ...

### Existence of a polynomial $Q$ of degree $\geq (p-1)/4$ in $\mathbb F_p[x]$ such that $QQ'$ factorizes into distinct linear factors
Such polynomials always exist, examples are Dickson polynomials of the first kind (with parameter $1$). For a positive integer $n$ these degree $n$ polynomials $D_n$ are most conveniently defined ...