# 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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### On $g^{2^x}=B$ modulo $p$

Let $p$ be prime and $g$ positive integer. Define $f(g,x)=g^{2^x} \bmod p$. Q1 Given $g,p,B=f(g,X)$, what is the complexity of finding $X$? If necessary assume $\varphi(p-1)$ is smooth. Some ...
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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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### How to count the number of tensors over a finite field of tensor rank $r$?

For simplicity, work over $\mathbb F_2$ and only consider order-$3$ equilateral tensors. For $r\in\mathbb Z_{>0}$, how many tensors $\mathfrak{T}\in\mathsf{Ten}_{n}^{\otimes 3}(\mathbb F_2)$ are ...
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### Bounds for the number of points on projective hyperelliptic curves over finite fields

Let $C$ be projective hyperelliptic curve over finite field $K$. What are bounds for the number of points $\#C(K)$? The Hasse-Weil bound requires smooth curves, and hyperelliptic curves are not smooth ...
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### Does there exist a polar decomposition of matrices over finite fields?

There exists a polar decomposition for matrices over the reals. What I would like to know is if an analog has been shown for groups of matrices over finite fields. If not, it would be great to get ...
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### Finite field special functions

I am looking for special class of involutive functions (f) with exactly one (zero) fixed point over finite field $F_q$ with properties: 1) $f(f(x))=x$ for any $x \in F_q$ , $f(x)=x$ iff $x=0$ 2) ...
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### Bivariate polynomial divisibility test of Spielman

Setup In his thesis (lemma 4.2.18, p. 97-98) Spielman describes a divisibility test for bivariate polynomials $E,P\in k[X,Y]$, where $k$ is a field (of positive characteristic for what I'm interested ...
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### A quantity associated to a field extension

Let $F\subset E$ be a field extension. So $E$ has a natural structure of $F$-vector space. A vector subspace $V\subset E$ is a special subspace if $F\subset V$ and $V$ is closed under the inverse ...
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### Lower bounds for class number of function fields with fixed $q$, growing $g$

Let $X$ be a smooth project curve of genus $g$ over the finite field with $q$ elements. Let $h$ be $\# \mathrm{Pic}^0(X)(\mathbb{F}_q)$. Weil showed that $h \geq (\sqrt{q}-1)^{2g}$. Lachaud and Martin-...
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### On solutions to linear system amalgam

Input: I. System of $\Omega(t)$ linear polynomials in $\mathbb F_2[x_1,\dots,x_{t}]$. II. System of $\Omega(t)$ linear polynomials in $\mathbb Z[x_1,\dots,x_{t}]$. Can we output a common $0/1$ ...
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### On a system of equations in $\mathbb F_2$

Input: System of $\Omega(t)$ independent polynomials in $\mathbb F_2[x_1,\dots,x_{t}]$ of degree $O(t)$. Can we output a common solution of the system in polynomial time? Can we output parity of the ...
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### Subspaces of vanishing permanent

Suppose that $p\ge 5$ is a prime, $n$ a positive integer divisible by $p-1$, and $L<\mathbb F_p^n$ a subspace of dimension $d=n/(p-1)$. Do there exist vectors $l_1,\dotsc,l_n\in L$ such that the ...
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### Properties of pointless projective curves over finite fields?

Probably not research level, feel free to downvote. We got construction of bounded degree projective curves with no points over finite fields. This construction generalizes to higher dimension. One of ...
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### Why the curve $x^2+y^2+y+1=0$ has only one point over $\mathbb{F}_{3^7}$?

According to both sagemath and Magma the curve $x^2+y^2+y+1=0$ has only one point over $\mathbb{F}_{3^7}$. The projective closure has only one point too. Q1 What hypothesis are missing to not violate ...
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### Chevalley-Warning-Ax for double covers

Let $f(x_1,\ldots,x_n)$ be a polynomial of degree $d$ with coefficients in the finite field $\mathbb F_q$ and let $V(f)\subseteq\mathbb F_q^n$ be its set of zeroes. Assume $d<n$. Then Chevalley ...
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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.
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### Low-Hamming weight vectors in low-dimensional subspaces of $\mathbb{F}_p^n$

What is the maximum number vectors of Hamming weight at most $w$ in a $d$-dimensional subspace of $\mathbb{F}_p^n$, where $w,d,p$ are constant and $p$ is odd. (The Hamming weight is the number of ...
### Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$?
If $E$ is a supersingular elliptic curve over $\mathbb{F}_{p^m}$ with $m\geq 2$ its endomorphism ring is a maximal order in a quaternion algebra ramified at $p$ and $\infty$ so there can't be a Weil ...
### On the real and finite field rank of a $0/1$ matrix - I
Let $M\in\{-1,0,+1\}^{n\times n}$ be a matrix of rank $r$. Consider the matrix $f(M)\in\{0,+1\}^{mn\times mn}$ where $0$ in $M$ is replaced by $m\times m$ all $0$ matrix, $+1$ in $M$ is replaced by \$m\...