# 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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### Gaussian integer square root solution [closed]

I'm trying to solve the equation $$E(x)=\frac{(1+ix)^{1/2}-(1-ix)^{1/2}}{(1+ix)^{1/2}+(1-ix)^{1/2}} \mod p,$$ where $p$ is a prime and $1+ix$ is a Gaussian integer over $p$. For some values of $x$, ...
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### Does Conway’s field of finite nim values have arithmetically tractable isomorphisms?

Conway constructed a field of characteristic 2 whose elements are the finite Nim values, indexed by the natural numbers. What is known about nontrivial automorphisms of this field? Do any of them ...
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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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### Supersingular curves over $\mathbb{F}_q$ and the splitting of $p$

I'm looking at chapter 4 of Waterhouse's "Abelian varieties over finite fields"; and Theorems 4.1 and 4.2 seem to use the following fact: Suppose that $E/\mathbb{F}_q$ is an elliptic curve ...
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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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### 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 ...
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### 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 ...
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### How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there?

How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there? As much as I have searched, I have not found any results that answer my question; not even for k = 1,2.
For a tournament $T$, let $\mathrm{dom}(T)$ be the order of a smallest dominating set in $T$. Let $q$ be a prime power congruent to 3 mod 4 and let $T_q$ be the Paley tournament on $q$ vertices. Is ...
### Question on rank of matrices over $\mathbb F_2$
$A$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $k\leq n-1$. $B$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $n$. $T$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $1$...