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# Tag Info

Accepted

Accepted

### 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 ...
Accepted

### 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 ...
Accepted

### 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 $x\mapsto x+x^{-1}$ ...
Accepted

Accepted

### Pointless groups

Perhaps I'm missing something, but it seems to me that $k=\mathbb{F}_2$, $G=\mathbb{G}_{m, k}$ works, where $H$ is the trivial subgroup.
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### Reinterpreting Galois descent over finite fields

Yes, assuming $X$ is reduced (so that its absolute Frobenius endomorphism is schematically dominant); this really is a formal consequence of the usual descent formalism (despite whatever red herring ...
It's $n(q-1)$. We must have $$\sum_{x_1,\dots,x_n \in \mathbb F_q}f(x_1,\dots,x_n)=f(0,\dots,0) \neq 0$$ but $$\sum_{x_1,\dots,x_n \in \mathbb F_q} \prod_i x_i^{e_i}$$ vanishes unless each $e_i$ ...
If $\# E(\mathbb{F}_p) = q$ and $j=0$, then the endomorphism ring is an order in the field of third roots of unity so $(p+1-q)^2 - 4p = -3u^2$ for some integer $u$. Now note that $(p+1-q)^2 - 4p$ is ...