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.
839 questions
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MMP over finite fields. State of the art
I would like to know how much of the results on the Minimal Model Programme (MMP) over fields of finite characteristicb which are usually only stated for varieties over algebraically closed fields, ...
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Checking if a supersingular elliptic curve shares an edge with the Spine
Let $E$ be supersingular elliptic curve defined over $\mathbb{F}_{p^2}$ and let $\ell$ be a prime. I'd like to know how one can check if E shares an edge, in the $\ell$-isogeny graph, with the set of ...
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Do isomorphic formal group laws over $\mathbf{F}_p$ have the same Frobenius characteristic polynomial?
Let $F(X,Y)$ and $G(X,Y)$ be formal group laws over $\mathbf{F}_p$ of finite height.
Hill proved (in this paper, Theorem E') that if $F$ and $G$ have the same characteristic polynomial of Frobenius, ...
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Solutions to $(x+y)(x^{-4^k}+y^{2^k})=1$ in the finite field $\mathrm{GF}(2^n)$
Let $F$ be the finite field ${\rm GF}(2^n)$ of order $2^n$. Let $X=F\setminus\{0\}$ be its multiplicative group. Suppose that $n/d$ is odd where $k\in\{1,\dots,n\}$ and $d=\gcd(k,n)$, so that the ...
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Mahler measure of monic irreducible polynomials over finite fields
Let $p$ be a prime. I thought of ways to select a unique monic irreducible polynomial of degree n over a finite field with p elements. Most things failed, but this here seems to give at most 2 such ...
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Fibre sizes for functions on finite fields
Let $F$ be the finite field ${\rm GF}(p^n)$ of order $p^n$. The automorphism $\theta(x)=x^{p^k}$ of $F$ has order $n/d$ where $d=\gcd(n,k)$. Consider the function $f(x)=\theta(x)-x^{-1}$ from $X:=F\...
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On a problem concerning Gauss sums over finite fields
Let $q=p^n$ be a prime power ($p$ is prime and $n\in\mathbb{Z}_{\ge 1}$) and $\mathbb{F}_q$ be a finite field with $q$ elements. Set $\widehat{\mathbb{F}_q^*}$ be the group of all multiplicative ...
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Hardness of comparing weight partitions of an affine space over $\mathbb{F}_2$
Let $A$ be an affine subspace of $\mathbb{F}_2^n$. Let $m\leq n$ and $Q_0, Q_1$ be linear maps $\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m$. Consider the following decision problem: Decide whether or not ...
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Families of matrices in $GF(2)$ whose linear combinations are full-rank
Consider the space of $n \times n$ matrices over the finite field $GF(2)$. Is it possible to choose $k$ matrices $A_i, i = 1 ,\cdots k$, such that for every non-zero vector $b \in \{0,1\}^k$ the ...
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On a claim of L. Denis about a Siegel Lemma in function fields
Let $\Omega$ be the completion of $\overline{\mathbb F_q\left(\left(\frac1T\right)\right)}$ for the topology induced by valuation $-\deg$. One denotes by $|.|_\Omega$ the associated normalized norm ($|...
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How to construct a matrix ($n\times m$, $m>n$) over $\mathbb{F}_2 $ with full rank, with the row width $m$ and entries' weight per row minimized?
How to construct a full-rank $n \times m$ matrix over $\mathbb{F}_2$ with $m > n$, minimizing width and row sparsity?
Goal:
Construct a matrix $X \in \mathbb{F}_2^{n \times m}$, with $m > n$, ...
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The most efficient algorithm for finding a root of a polynomial over finite field
I have a polynomial of degree around $2^{35}$ over a finite field $\mathbb{F}_p$, where $p$ is a 64-bit prime number.
What is the most efficient algorithm for finding a root of such a polynomial? (...
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Non-concentration implies full rank of random matrix over large field?
Consider a random $2 \times n$ matrix $X$ whose entries $X_{ij}$ take values in a finite field $E$ of characteristic $p$. Although the entries may not be independent, they satisfy the following non-...
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Bounds for a 2D hyper Kloosterman sum
Let $a,b,c,q$ be positive integers. One way to generalize the standard Kloosterman sum to two variables is
$$
K(a,b,c;q) := \sum_{\substack{x_1\, \text{(mod $q$)}\\ (x_1,q)=1}} \sum_{\substack{x_2\, \...
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An invertibility property of submatrices of a matrix
Let $\mathbb{F}_n$ denote the finite field with $n$ elements.
Suppose that the (non-tall) matrix ${\bf M} \in \mathbb{F}_n^{r \times n}$, where $r \leq n$, has rank $r$ and for any $k \leq r$, the ...
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Is there closed form for the factorization of $F(x)$ of special form modulo $N$?
We got a fast probabilistic algorithm for factorizations of univariate
polynomials of special form modulo $N$ and would like to know if it
is known or trivial.
Let $N$ be composite integer with ...
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Solvability of F_p-points of semisimple groups
Let $G$ be a semisimple algebraic group over the finite field $\mathbf{F}_p$.
Question 1. When is $G(\mathbf{F}_p)$ a solvable group?
The obvious guess is "rarely". Due to some well-known ...
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Is there a proof of Wedderburn's and Artin–Zorn's theorems on finite division algebras using only synthetic projective geometry?
In this post, I don't require division algebras to be necessarily associative (but I do require a unit element). In synthetic projective geometry (part of incidence geometry), there is the notion of a ...
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Number of roots of a quadratic form over GF(2)
If $Q(x) = x^T A x$ with $x \in GF(2)^n$ and $A \in GF(2)^{n \times n}$, is there a way to find how many roots $Q(x)$ has based on any properties of $A$ (e.g., rank, etc.)?
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Average number of $\mathbb{F}_p$-points over twists of a variety
Let $p \gg 1$ be a sufficiently large prime. I recently stumbled across a fascinating fact about the number of $\mathbb{F}_p$-points on elliptic curves over finite fields. Specifically, we have:
Fact ...
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Uniqueness of differences of roots of polynomials over finite field
Let $f$ be a polynomial over a finite field $\mathbf{F}_p$ with $p \neq 2$. Let $R$ be the roots of $f$ in some extension field. I am interested in the multiset of differences $R - R = \{ r - s \mid r,...
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Distinct eigenvalues of random matrix over finite field
Let $A$ be a uniformly random matrix in $\mathrm{M}_n(\mathbf{F}_p)$.
It is well known that, as $p$ is fixed and $n$ tends to infinity, $A$ has repeated eigenvalues (over the algebraic closure $\...
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Centralizer of PSL in PGL and of SL in GL: reference request
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\PSL{PSL}$Consider the general linear group $\GL(n,q)$ over a finite field with $q$ elements and ...
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Asymptotics of A000613
The general linear group $GL_n(\mathbb{F}_2)$ acts on the powerset $2^{{\mathbb{F}_2}^n \setminus \{0\}}$ by multiplication: $A \cdot S := \{Ax \in {\mathbb{F}_2}^n : \, x \in S\}$, for an invertible ...
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Ramanujan graph element in $\mathsf{PSL}(2, \mathbb{Z}_q)$
I am trying to follow the construction of the Ramanujan graph $X^{p, q}$ given in the paper 1. The first few steps of the construction proceed as follows:
Let $p$, $q$ be two unequal primes that are ...
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