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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Does every geometric progression contain a small remainder modulo a large prime?
The exact question I am interested in is the following.
Fix a small $\varepsilon\in(0,1)$ and an integer $q\ge 2$ (you may assume that $q$ is prime if it helps though I believe it shouldn't matter too ...
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Maximum probability of a set of vectors from $ \mathbb{F}_2^n $ being linearly independent
Suppose $ m $ vectors from the vector space $ \mathbb{F}_2^n $ are selected independently according to a distribution $ P $ over $ \mathbb{F}_2^n $. Here $ \mathbb{F}_2 $ denotes the field with two ...
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Injectivity of pushforward of rational Chow groups
I'd like to know whether there is a known counter-example to the following statement. Let $X$ be a smooth projective variety over a finite field. Let $Z$ be a codimension $2$ smooth subvariety which ...
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Computing monodromy groups of curves over function fields
Suppose I consider a hyperelliptic curve given by an equation such as $y^2 = x^{n} + tx + 1$ or some variation on this (where $t$ is a parameter on $\mathbb P^1$ and this curve is really a surface ...
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Reference for integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$
I was looking for a reference which discusses the structure of finite integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$. In particular, I am interested in understanding what the abelian group of its ...
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Related involutions
Let's say that we have finite field $\mathbb F_q$ and we have a couple of involutions $g,f$ with exactly one fixed point (zero).
Let's take any element $\alpha \in \mathbb F_q$
Let's start applying ...
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invariant subspaces of general linear groups for finite fields
Let $K$ be a finite field, let $n\ge 1$ be an integer and let $G=\mathrm{GL}(n,K)$ be acting linearly on a finite dimensional $K$-vector space $V$. Although $G$ is a reductive group, it is not ...
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Loewy structure of $S_4$
How to deduce the Loewy Structure of $kS_4$ where $k$ has characteristic 2. I can compute the Cartan matrix and Decomposition matrix with Brauer Characters without difficulties. But when it comes to ...
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Standard interpretation of permanents (of orthogonal included) over finite fields
Given a $0/1$ matrix in $\mathbb Z^{n\times n}$ the standard interpretation of permanent of the matrix is the number of perfect matchings in the underlying $2n$ vertex balanced bipartite graph with ...
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Invertible matrix by using polynomial in LDPC codes
I am studying about QC-LDPC codes.
These codes can be represented by matrices or polynomial.
For instance:
Example.
So, we have two polynomials: $a_1(x) = 1 +x$ and $a_2(x) = 1+x^2+x^4$.
The second ...
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Polynomial form/Fourier transform of rational function over finite affine space
I am certainly going to make a mess of any serious algebraic terminology, so bear with me as I present my problem arising from a probability problem.
Consider the space of sequences of $n$ zero-one ...
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On some rational points on an elliptic curve over finite field
Let $p\equiv3\pmod4$ be a prime. We consider the elliptic curve $E$ over the finite field $\mathbb{F}_p$
(in affine coordinates) defined by
$$y^2=x^3+x.$$
Clearly the discriminant of $E$ is $-2^6$. ...
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Is finite field version kakeya conjecture still true when changing the line of every direction with only 2(or several but not the full line)element?
The classical finite field Kakeya conjecture state as following (for conveinent, all version of kakeya conjecture is state in hasdorff dimension version):
$\mathrm{Finite\ Field\ Kakeya\ ...
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Do you know which is the minimal local ring that is not isomorphic to its opposite?
The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.
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Explicit large finite fields in characteristic $2$
Every finite field of characteristic $2$ ist given by $\mathbb{F}_2[x]/P(x)$ for some irreducible polynomial $P\in \mathbb{F}_2[x]$.
For small degree, a simple algorithm gives a way to find $P$. Is ...
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Symmetric matrices of hyperbolic and elliptic type with certain kind of trace zero
I have been working on a problem related to determinantal varieties in symmetric matrices. I am stuck at the following point and would like to get some reference/help for the following question.
Let $\...
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Dyadic models in number theory and "spillover"
In a classic blog post, Tao discusses the appearance of "dyadic models" in various guises in various areas of math. The number-theoretic version of the idea is to study polynomials over a ...
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Discrete logarithm for polynomials
Let $p$ be a fixed small prime (I'm particularly interested in $p = 2$), and let $Q, R \in \mathbb{F}_p[X]$ be polynomials.
Consider the problem of determining the set of $n \in \mathbb{N}$ such that $...
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Orbit counting polynomials over finite fields
Let $X$ be an affine variety defined over $\mathbb{Z}$ and let $G$ be an algebraic group defined over $\mathbb{Z}$. Let $q$ be a power of a prime number. We write $\mathbb{F}_q$ for the field with $q$ ...
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Richardson varieties over finite fields
Let me start with some background to set the notation before I ask my question.
Let G be a semisimple algebraic group over some algebraically closed field K, and suppose we have fixed a Borel ...
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The period of Fibonacci numbers over finite fields
I stumbled upon these very nice looking notes by Brian Lawrence on the period of the Fibonacci numbers over finite fields. In them, he shows that the period of the Fibonacci sequence over $\mathbb{F}...
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Permutation induced by multiplication of finite field elements [closed]
Consider a finite field $\mathbb F$. Let $a \in \mathbb F$. Then multiplication by $a$ induces a permutation on the field elements. $0 \rightarrow 0$, $1 \rightarrow a$, $2 \rightarrow 2a$, etc.
Is ...
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$\delta$-equidistributed polynomials over finite fields
I'm trying to show that a polynomial over finite (prime) field is "close enough" to being equidistributed over its range. A polynomial $p(\cdot)$ from $\mathbb{F}^n$ to $\mathbb{F}$ is $\...
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Chains of numbers generated by 2 involutions
$\DeclareMathOperator\GF{GF}$Consider the finite field $\GF(p)$ for prime $p$.
Consider the pair of involutions $f(x) = 1-x$ , $g(x) = 1/x$, and the chain of numbers generated by these 2 involutions ...
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Counting the number of linearly independent primitive elements of a field extension
Let $L/K$ be a field extension with extension degree $n>1$. We say $L/K$ is simple if $L=K(a)$ for some a in $L$. In this case, $a$ is called a primitive element. My question is now about the ...
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When are Hamming codes cyclic?
I've asked this question on math.stackexchange before, but it has not been solved.
The following statement appears to be true:
The $q$-ary Hamming code of codimension $r$ over $\mathbb{F}_q$ is ...
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Arrangement of subspaces over finite fields
I'm trying to find out what is already known about the following setup.
Let $V$ be an $n$-dimensional vector space over a finite field $F_q$ (I'm mostly interested in the case where $q$ is prime), and ...
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Enumerating subspaces of $\mathbb{F}_q^n$ in terms of words and inversions
When $q$ is a prime power, then on the one hand the $q$-binomial coefficient $\binom{n}{k}_q$ equals the number of $k$-dimensional subspaces of $\mathbb{F}_q^n$, and on the other hand it is the ...
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computer algebra system for polynomial algebras over finite fields
Is there a computer algebra system that can do arithmetic over polynomial algebras over finite fields where I can specify the extension?
Exempli gratia, if $f(x), g(x) \in \mathbb{F}_p[\mu]/(m(\mu))[...
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Fibonacci-like sequences in $\mathbb{F}_q$ where each element only depends on the previous one
Given a prime power $q$, consider all sequences $(a_n)_{n\in\mathbb{Z}}$ in $\mathbb{F}_q$ for which $a_{n+1}=a_n+a_{n-1}$ for all $n\in\mathbb{Z}$. Call such a sequence simple if there exists a ...
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Integral points on elliptic curve and the Lee norm
This question is based on small experiments I have done in Sagemath and if it is not research level, I will move it to MSE:
Let $E$ be an elliptic curve defined with coefficients in $\mathbb{Z}$.
The ...
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Centralizers of $\mathbb{F}_q$-rational semisimple elements of a finite group of Lie type
Let $\mathbb{G}$ be a connected reductive $\mathbb{F}_q$ algebraic group over its algebraic closure $\bar{\mathbb{F}_q}$, and $\mathbb{T}$ be an $\mathbb{F}_q$-defined maximal torus. Let $\Phi$ be the ...
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Enumeration of field generators of a finite field over $\mathbb{F}_{q}$, which are $m^{th}$ powers in the same field
Consider the finite field extension $\mathbb{F}_{{q}^{d}}$ over $\mathbb{F}_{q}$, where $q=p^a$ for some prime $p$. We assume $d\geq 2$. Let,
$$ S=\{ \alpha \in \mathbb{F}_{q^d}\hspace{0.1 cm} | \...
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How many Lie and associative algebras over a finite field are there?
This question is related to the following general question:
Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, ...
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The size of endomorphism rings and the relation to ordinariness of Abelian surfaces
For Elliptic curves over a finite field, there is a very useful characterization of ordinary elliptic as those with commutative, quadratic endomorphism rings and of supersingular curves as those with ...
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System of linear equations in positive characteristic
Let $K$ be a field of positive characteristic $p$. Consider the system of $\mathbb F_p$-linear equations
$$\left\{\begin{array}{ccl}
a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n&=&b_1\\
a_{11}x^p_1+a_{...
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Reference request concerning splitting fields for groups that are related to special symmetric groups
Denote the symmetric group of order $n!$ by $S_n$. Let $H:=S_p$ for an odd prime $p$.
Every finite field $k$ is a splitting field $(^*)$ for $kH$, in particular $k:=\mathbb{F}_p$.
Questions:
Is $k:=\...
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Bijections between binary sequences and primitive elements in a finite field [duplicate]
Let $n>1$ be a natural number. We call a binary sequence $(b_1,\ldots, b_n)\in \{0,1\}^n$ $rigid$ if it is not a proper power of a sequence of shorter length. So for example $(0,1,0,1) = (0,1)^2$ ...
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Image of a polynomial function $x^2+y^2-x+y-axy$ over $\mathbb{F}_p$
Let $p$ be an odd prime and $h(x)=x^2+ax+1$ be an irreducible polynomial over the field $\mathbb{F}_p$. I need to prove that the function
$$\Psi: \mathbb{F}_p^2 \longrightarrow \mathbb{F}_p, \quad (x,...
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Polynomial representation of modular arithmetic in finite fields
Let $n \in \mathbb{N}$ be a predefined integer. Consider the following bijection (between the ring of integers modulo $2^n$ and finite field with $2^n$ elements:
$$ \phi: \mathbb{Z}_{2^n} \to \mathbb{...
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Maximum pairwise-product of a set in $Z_p$
Let $A$ be a subset of $[1,p-1]$ of size $N$, for a prime $p$.
My question is what is the most efficient algorithm to find: $$\max \{(x \cdot x')~mod~p~|~ x,x'\in A\}$$
In other words, how efficiently ...
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Fast double exponentiation in finite fields
Let $p$ be a prime, and let $\mathbb{F}_p$ be the finite field with $p$ elements. Let $a$ be a non-zero element of $\mathbb{F}_p$. Can we quickly evaluate $a^{2^r} \mod{p}$? Using repeated squaring, ...
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Cyclic codes: sparse codewords not orthogonal to the all-ones vector
Is it true that for any sufficiently large prime $p$, there exists a prime $q\ne p$ and a cyclic code of length $p$ over $\mathbb{F}_q$ that contains a codeword of Hamming weight at most ord$_p(q)$ ...
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Simultaneous similarity of matrices over finite fields
Suppose $A,B\in SL(3,F_q)$, where $F_q$ is the finite field of order $q$ and $SL(3,F_q)$, the group of matrices with determinant one and entries from $F_q$ , are such that $A$ has eigenvalues in $F_q$...
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Pointless, non-singular, absolutely irreducible affine plane curves over finite fields
We think the following is true:
For all sufficiently large primes $p$ and all natural $g \ge 1$, there
exists affine plane curve $f(x,y)=0$ over $\mathbb{F}_p$ which
is non-singular, absolutely ...
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Number of solutions of a degree 4 polynomial equation over a finite field
Suppose that $q$ is a prime power and $\xi, \eta\in \mathbb{F}_q$ are nonzero. A computer calculation for $q<70$ suggests that the number $N$ of $4$-tuples
$(a,b,c,d)\in\mathbb{F}_q^{4}$ satisfying ...
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Intrinsic characterisation of a class of rings
This may be well known, but I was unable to find an answer browsing literature. Let us temporarily call a commutative (unital) ring $R$ an O-ring if there exists an integer $n \ge 1$, a local field of ...
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Solutions to system of polynomial equations over finite fields
If $P_1$, $P_2$, ..., $P_m$ are $n$-variate homogeneous polynomials of degree d over a finite field $F_q$, where $q$ is much larger than $d$, but much smaller than $n$, then do we know good lower ...
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Subgroups of multiplicative groups of the finite field with Mersenne prime order
I have a question about properties of the multiplicative groups.
Let we have finite field of prime order $2^k$ -1. $k$ is co-prime with $$\frac{2^k-2}{k}$$.
It is clear that multiplicative group of ...
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Fantastic properties of Z/2Z
Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...
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