# Questions tagged [finite-fields]

A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.

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### Varieties with the same number of $\mathbb{F}_p$-points for all but finitely many primes

If two varieties over $\mathbb{Q}$ have the same number of $\mathbb{F}_p$-points for all but finitely many primes do they have the same number of $\mathbb{F}_{p^n}$-points for all $n>1$ and for all ...
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### Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?

It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G. More generally here (MO275769) Qiaochu Yuan ...
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### What is a function field analog of Giuga's conjecture?

Giuga's conjecture (1950), which is still open and has strong numerical support, reads : Let $n$ be a positive integer. If $1+\sum_{k=1}^{n-1}k^{n-1} \equiv 0\pmod{n}$ then $n$ is prime. What would ...
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### Polynomials which are functionally equivalent over finite fields

Recall that two polynomials over a finite field are not necessarily considered equal, even if they evaluate to the same value at every point. For example, suppose $f(x) = x^2 + x + 1$ and $g(x) = 1$. ...
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### Galois group of zeta function of hyperelliptic curve

Let $f \in \mathbb F_q[T]$ be monic, squarefree. Can we say anything on the Galois group of $Z_f$, the zeta function of the hyperelliptic curve $y^2=f$, directly in terms of $f$ (coefficients or ...
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### How large can the dimension of a 'Span of powers of a finite field basis' be?

Let $p$ be a prime. For finite field $\mathbb{F}_{p^k}$ and $d\in\mathbb{Z}^+$, I am considering the following quantity, where we interpret the field $\mathbb{F}_{p^k}$ also as a $\mathbb{F}_p$-vector ...
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### Number of involutions in finite reductive groups

Let $G$ be a connected split reductive group over $\mathbb{Z}$. Let $n$ be a positive integer. Let $i_n(q)$ be the number of elements of $G(\mathbb{F}_q)$ satisfying $x^n=1$. Question: Is there a &...
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### p-adic logarithms with fixed precision

Probably this is easy, but we would like to see it on paper. Let $p$ be prime and $D,g,n$ positive integers. Let $A=g^n \bmod p^D$. Let $\log(p,a,D)$ be the p-adic logarithm with precision $D$. In ...
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### Isomorphic endomorphism algebras implies isogenous (for abelian varieties over finite fields)?

$\newcommand{\F}{\mathbb{F}} \newcommand{\End}{\mathrm{End}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}}$ I would like to know if the following is true: Proposition A : Let $A_1, A_2$ ...
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### On the real and finite field rank of a $0/1$ matrix - II

Let $M\in\{-\ell,\dots,-1,0,+1,\dots,+\ell\}^{n\times n}$ be a matrix of rank $r$ where $\ell\geq1$ such that there is a permutation matrix in $\{0,1\}^{m\times m}$ of order $2\ell$. Fix a permutation ...
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### Least prime in Artin's primitive root conjecture

Let $a$ be an integer which is neither a square nor $-1$. Artin's conjecture states that there are infinitely many primes $p$ for which $a$ is a primitive root modulo $p$. My question is whether there ...
### On a matrix problem in the field $\mathbb F_2$
Given $M$ a symmetric matrix in $\mathbb F_2^{n\times n}$ having $\mathsf{det}_\mathbb R(M)\neq0$ (non-singular in reals) and satisfying $PMP'=(M+J+I)$ or $P(M+J+I)P'=M$ where $P$ is a permutation ...