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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A Vandermonde-type system
For a prime $p$ and $a_1,\dotsc,a_n\in\mathbb F_p^\times$, consider the system of equations
$$ \begin{cases}
\begin{align}
a_1 + \dotsb + a_n &= 0 \\
a_1x_1 + \dotsb + a_nx_n &...
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Nature of polynomials of the form $x^n-a$ over finite fields
I state the following theorem from Serge Lang's Book- Algebra(3rd edition).
Theorem: Let $k$ be a field and $n$ an integer $\geq$ 2. Let $a\in k, a\neq 0$. Assume that for all primes $p$ such that $...
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Upper bounds on the order of a number in prime fields
Let $k$ be a fixed integer and for any prime number $p$ larger than $k$, let $\text{Order}(k,p)$ be the order of $k$ in $\mathbb{F}_p$ (i.e., $\text{Order}(k,p)$ is the least integer $n$ such that $k^...
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Imagining linear maps between finite fields
I can't imagine a right picture of a linear transformation $\mathbb{F}_{p} \mapsto \mathbb{F}_p$ or $\mathbb{F}_{p^2} \rightarrow \mathbb{F}_{p^2}$ etc (over the field $\mathbb{F}_p$) although they ...
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1/2 Wilson's theorem
During my research I came across this question :
Question: What's the value of $x_p=(\dfrac{p-1}{2})! \mod p$ when $p>3$ is prime ?
Remark: It's easy to see $x_p^2 \mod p=(-1)^{\dfrac{p+1}{2}} \...
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Satake correspondence for groups over finite field
I asked the same question in MSE, but I didn't get any answer. So I decided to post it here, too.
In Langlands' program, Satake correspondence gives a correspondence between unramified ...
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Smoothness of the moduli space of Drinfeld modules
I'm studying the proof of Thm 1.5.1. in Laumon's "Cohomology of Drinfeld Modular Varieties". Notation: $\mathfrak{m}$ is a square zero ideal of $\mathcal{O}$ and $k=\mathcal{O}/\mathfrak{m}$. Laumon ...
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Most points on a degree $p$ hypersurface?
Let $p$ be a prime. Let $f \in \mathbb{F}_p[x_1, \ldots, x_n]$ be a homogenous polynomial of degree $p$. Can $f$ have more than $(1-p^{-1}+p^{-2}) p^n$ zeroes in $\mathbb{F}_p^n$?
Basic observations: ...
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What is the minimum $k$ such that $A^k \equiv I$ mod p for invertible matrices?
Let $F$ be a finite field of order $p$, where $p$ is prime. For any $n\times n$ matrix $A$ that is invertible over $F$, then there would appear to exist integers $k$ such that $A^{k} = I$. My question ...
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On equality of two quotients of a congruence subgroup
Related question: Non-torsion part of the abelianisation of congruence subgroups
Let $A = \mathbb{F}_q[T]$ be the ring of polynomials with coefficients in a finite field, with $N$ a nonconstant ...
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Is there an easy way to compute the maximum isotropic subspace over finite fields?
Given a quadratic form (or a symmetric $n \times n$ matrix $A$), an isotropic subspace is a subspace $U$ such that $$U^t A U=0,$$
If I am not mistaken, when the matrix is over reals, the maximum ...
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The slope of $nx\ \%\ m$ [closed]
(This is a follow-up question to another question I asked at MSE. I edited due to an important hint from Will Sawin - see his comments below.)
There will be this question at the end of this post:
...
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Groups implementable by finite field
I'm interested in finding all groups for which the group operation (and inverse map) may be implemented using finite field arithmetic.
I've done some searching and have come across "algebraic groups",...
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A summation of powers defined by an equation over finite fields
Let $p$ be an odd prime and let $k\mid q$ for some positive integer numbers $k$ and $q$. Suppose that $r \in \mathbb{F}_{p^q}$ has multiplicative order $p^k-1$.
For each $1\leq u \leq p^k-3$, the ...
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Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?
$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons.
...
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On an exercise in The Probabilistic Method : random dilate of a set in a finite field
This is related to Problem $4.6$ in ``The Probabilistic Method'' by Alon and Spencer, where one essentially has to prove the following:
Let $p$ be a prime, and $A$ be any subset of $\mathbb{F}_p$. ...
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Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for finite fields?
The Balazard-Saias-Yor criterion for the Riemann Hypothesis states that the latter is equivalent to the statement that
$$\int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s^2|}|ds|=0$$ where $\zeta$ denotes ...
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A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (II)
As in Question 319254, for an odd prime $p$ and integers $c,d$ we let
$$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right).$$ If $p\equiv1\pmod4$, then we obviously have
\begin{align}&\...
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Write the algebra closure of $F_p$ as union of finite fields [closed]
This question follows Field theory by Steven Roman, Chapter 9, Exercise 20.
Denote the algebraic closure of the finite field $F_q$ by $\Gamma(q)$, and let $a_n$ be any strictly increasing infinite ...
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Permutations of squares and finite fields
Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$, and let
$$S(n)=\bigg\{\sum_{k=1}^nk^2\pi(k)^2:\ \pi\in S_n\}.$$
Motivated by Question 316142 of mine, here I ask the following ...
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Covering the finite plane with lines
This is, essentially, a geometrically rendered version of the question I asked a week ago, with the emphases slightly shifted; it seems more natural and appealing (to me, at least) in this form.
Let ...
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On the set $\{\sum_{k=1}^n \lambda_ka_k:\ a_1,\ldots,a_k\ \text{are distinct elements of}\ A\}$
For a field $F$ let $p(F)=p$ if the characteristic of $F$ is a prime $p$, and $p(F)=+\infty$ if $F$ is of characteristic zero.
In 2007 I considered the linear extension of the Erdos-Heilbronn ...
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Non-torsion part of the abelianisation of congruence subgroups
I've posted this question on math.stackexchange, but haven't gotten any responses so I'm trying here instead.
Let $A = F_q[T]$ be the ring of polynomials in one variable with coefficients in a finite ...
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Submersion implies many rational points in image?
Let $A \colon V \to W$ be a surjective linear map
(defined over $\mathbb{Z}$),
inducing a projection
$\alpha \colon \mathbb{P}(V) \to \mathbb{P}(W)$.
Let $X \subseteq \mathbb{P}(V)$ and $Y \...
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The growth of class number in $\mathbb{Z}_p$-extensions of function fields
Let $X$ be a curve (proper, smooth, ...) over a finite field $\mathbb F_q$ where $q$. Suppose also that $\mathbb F_q$ contains the $p$-th roots of unity, in this case we have the following (unique) ...
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Polynomials passing through points with tangential conditions
In corollary here http://math.mit.edu/~lguth/Exposition/erdossurvey.pdf on polynomial methods it is said "(Parameter counting) If $S\subset\mathbb F^n$ is a finite set, then there is a non-zero ...
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$\mathbb{F}_q$-rational elements in unipotent classes of a finite group of Lie-type
I've tried posting this question on MSE, but didn't manage to get an answer there, so I'm trying again here. Sorry in advance if this question is trivial or trivially false. I haven't managed to find ...
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The minimum rank of a matrix over GF(2) when part of non-zero off-diagonal elements are set to be zero
Given an $n\times n$ matrix $A$, whose elements are over $GF\left(2\right)$ and all diagonal elements are $1$. There are $m\ (m\leq n^2-n)$ non-zero off-diagonal elements in $A$. If we are allowed to ...
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2-Torsion in Jacobians of Curves Over Finite Fields
Let $C$ be a (smooth, projective) curve over a finite field $\mathbb{F}_q$, and let $J_C(\mathbb{F}_q)$ denote its Jacobian. Suppose the genus $g$ of $C$ is at least $1$.
Question 1: Are there curves ...
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Factorisation of polynomials over finite field
Is there a method to factorise a polynomial, for $k \leq m$ and $a_i \in \mathbb{F}_p$,
$$
1 + t^k(1 + a_1 t + a_2 t + \ldots + a_m t^m)^k
$$
as a product $$
(1 + t^k)^{x_1} \cdots (1 + t^l)^{x_l} \...
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Roots of lacunary polynomials over a finite field
If $P$ is a polynomial over the field $\mathbb F_q$ of degree at most $q-2$ with $k$ nonzero coefficients, then $P$ has at most $(1-1/k)(q-1)$ distinct nonzero roots.
Does this fact have any standard ...
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Smallest set of nonzero vectors in $\mathbb F_2^n$ which intersects every 2-dimensional subspace
What is the smallest set of nonzero vectors in $\mathbb F_2^n$ which intersects every 2-dimensional subspace?
For example, for n = 3, the set {001, 010, 011} does the job, and is minimal. For n = 4, {...
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Lifting Lang-Steinberg to DVR's in Characteristic 0
Let $A$ be a compact DVR in characteristic $0$, uniformizer $\pi$ and residue field $k$. Let $A\subset B$ be a complete DVR with the same uniformizer $\pi$ and algebraicly closed residue field $F$. ...
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Intersection of two varieties in $\mathbf{F}_q^n$
Suppose we identify $\mathbf{F}_q^n$ with $\mathbf{F}_{q^n}$. Let $X_n$ be the irreducible hypersurface defined by $Nx=1$ where $N$ is the norm map. There is an analogous hypersurface $X_{n-1}$ in $\...
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Upperbounding a sum of Legendre-Symbols
Let $p$ be a prime with $p\equiv 3 \mod 4$, for any $\mathcal{I} \subset \lbrace 0,...,p-1 \rbrace $ and any $\mathcal{J} \subset \lbrace 0,...,p-1 \rbrace $ with $\vert\mathcal{I}\vert \leq \sqrt{p} $...
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Are the integers a vector space or algebra over "some" field or over "some" ring?
Every vector $v$ in a finite-dimensional vector space space $V$ of dimension $n$ over a field $F$ has a unique representation in terms of a basis ${\frak B} \subseteq V$, where a basis for $V$ is a ...
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Monoid cohomology of $\mathbb{N}$ for a linear algebraic group
Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...
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Surjectivity of norm map on subspaces of finite fields
It is basic that the norm map $N:\mathbf{F}_{q^n}^* \to \mathbf{F}_q^*$ is surjective for finite fields. In fact $N(x) = x^{(q^n-1)/(q-1)}$. How well does this simple fact extend to subspaces?
A ...
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Decomposition of a Matrix by Sparse Matrices
Let $\mathbb{F}$ is a field. Consider an $n \times n$ matrix $\bf A$ over $\mathbb{F}$.
$\bf A$ is called sparse matrix over $\mathbb{F}$ iff the number of non-zero entities of $\bf A$ be at most ...
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Notions of convergence over extensions of finite fields
Let $\displaystyle Q_p[x] = \left\{\frac{p(x)}{q(x)} \mid \, p(x),q(x) \in \mathbb{F}_p[x], \, q(x) \neq 0 \right\}$ denote the field of fractions extending $\mathbb{F}_p[x]$. If we consider the ...
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Solving system of multivariable algebraic equations over $\mathbb Q$ by reducing over $\mathbb F_p$
I try to solve the finite system of multivariable algebraic equations with coefficients from $\mathbb Q$. It would be sufficient for me to prove that there is only finite number of solutions over $\...
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Pair of vectors multiplied by a random matrix and its inverse transpose are distributed randomly up to their dot product
Given arbitrary nonzero vectors $\vec{x}_1, \vec{y}_1, \vec{x}_2, \vec{y}_2 \in \mathbb{Z}^{n}_p$ ($p$ prime) with $\langle x_1, y_1 \rangle = \langle x_2, y_2 \rangle$, I am trying to show that: $(...
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Reference request: Number of elliptic and hyperbolic quadratic forms of a given rank over a finite field
My question is over the finite field $\mathbf{F}_q$ of $q$ elements. It is well known that a symmetric matrix of odd rank corresponds to a parabolic quadratic form but even rank symmetric matrices ...
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2
answers
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Alternate descriptions of finite fields
The finite field of order $p^n$ is isomorphic to $(\mathbb Z/p \mathbb Z)[X]/(P)$, where $P$ is an irreducible polynomial in $(\mathbb Z/p \mathbb Z)[X]$ of degree $n$. This describes every finite ...
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Lacunary fully reducible polynomial over a finite field
The following problem is motivated by this MO question on rich directions determined by a set of a finite plane.
Problem Does there exist a constant $C$ such that for all odd primes $p$ there is a ...
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Balancing points with lines
$\newcommand{\F}{\mathbb F}$
Suppose that $p$ is a prime, and $k<p/2$ a positive integer.
Consider a system of $k$ distinct directions in the affine plane $\F_p^2$, and the system of $k$ pencils ...
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On primitive roots of the form $5^k+10^m$ with $k$ and $m$ nonnegative integers
Let $p$ be any prime. It is well known that the set
$$G_p=\{0<g<p:\ g\ \text{is a primitive root modulo}\ p\}$$
has cardinality $\varphi(p-1)$, where $\varphi$ is Euler's totient function. It is ...
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If number of points on a manifold is $q^n ( [n+1]_q )$ does it imply a geometric relation to $A^n (P^n)$?
Consider an algebraic manifold whose number of points is $q^n ([n+1]_q)$. Is there a geometric relation to $A^n (P^n)$? In particular, is there an equivalence in the Grothendieck ring of varieties ...
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When spreading out a scheme, does the choice of max ideal matter?
I'm looking at Serre's paper How to use Finite Fields for Problems Concerning Infinite Fields. Specifically I'm trying to use the techniques in the proof of Theorem 1.2 to write out the details of the ...
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Linear permutations commuting with $x\rightarrow x^{-1}$
Let $F = \operatorname{GF}(2^n)$ be a finite field. Define a permutation $\phi:F \rightarrow F$ by the formula
$$
\phi(x) = x^{-1}, \ x\neq 0; \ \phi(0) =0.
$$
We say that a permutations $\psi$ of $F$ ...
