# 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.

69
questions

**13**

votes

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### Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?

Question 1 What is the number of pairs of commuting elements in GL_n(F_q) ?
I am aware of many results concerning commuting elements in Mat_n(F_q), but I am interested in GL i.e. non-degenerate ...

**67**

votes

**6**answers

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### How to recognise that the polynomial method might work

A couple of days ago I was at a nice seminar given by Christian Reiher, during which he told us about a short proof of the following special case of a theorem of Olson.
Theorem. Let $(a_1,b_1),\dots,(...

**53**

votes

**14**answers

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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 ...

**32**

votes

**9**answers

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### How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings,
fields, graphs, partial orders, etc.
...

**20**

votes

**4**answers

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### Exhibit an explicit bijection between irreducible polynomials over finite fields and Lyndon words.

Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree $n$...

**21**

votes

**1**answer

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### maximal order of elements in GL(n,p)

I am looking for a formula for the maximal order of an element in the group $\operatorname{GL}\left(n,p\right)$, where $ p$ is prime.
I recall seeing such a formula in a paper from the mid- or early ...

**18**

votes

**1**answer

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### Conjecture: The number of points modulo $p$ of certain elliptic curve is $p$ or $p+2$ for $p$ of form $p=27a^2+27a+7$

Numerical evidence suggests a conjecture that the number of
points of certain elliptic curve over $\mathbb{F}_p$ is
either $p$ or $p+2$ for $p$ of certain form.
Let $p$ be prime of the form $p=27a^2+...

**6**

votes

**1**answer

238 views

### 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 ...

**15**

votes

**5**answers

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### Good source for representation of GL(n) over finite fields?

I'd like to gain some understanding of unitary representations of GL(n) over finite fields. Any good source would be appreciated.
======== edit =========
My original question was ambiguous. ...

**10**

votes

**1**answer

630 views

### 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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votes

**1**answer

227 views

### Primes of the form $p=3a^2+3ab+b^2$ or $p=27a^2+27ab+7b^2$ and the number of points of $y^2=x^3+2$ modulo $p$

Trying to generalize this answered question based on limited numerical evidence.
Let $E / \mathbb{F}_p : y^2=x^3+2$.
Conjecture 1 Let $p=3a^2+3ab_0+b_0^2$ be prime and $a,b_0$
positive integers.
...

**12**

votes

**3**answers

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### Quadratic reciprocity and Weil reciprocity theorem

I was told that Weil reciprocity theorem (one has two meromorphic function $f,g$ on a complex curve $C$, so $\prod\limits_{x\in C} g(x)^{ord_xf}=\prod\limits_{x\in C}f(x)^{ord_xg} \ $ where $ord_xf$ ...

**19**

votes

**4**answers

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### Non-split extensions of $GL_n(F_q)$ by $F_q^n$ ?

A very naive question :
I just learned that there is a non-split extension of $GL_3(F_2)$ by $F_2^3$ (with standard action). It can be realized as the subgroup of the automorphism group $G_2$ of ...

**9**

votes

**2**answers

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### What is an example of a smooth variety over a finite field F_p which does not lift to Z_p?

Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.

**16**

votes

**3**answers

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### Periodic orbits and polynomials

There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system.
fact 1 Consider the "tent map" f:[0,1]→[...

**14**

votes

**3**answers

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### Probability of coprime polynomials

Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let
$f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...

**16**

votes

**2**answers

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### Examples of problems where considering "discrete analogues" has provided insight or led to a solution of the original problem

The Kakeya conjecture posits that any Kakeya set in $\mathbb{R}^n$ has dimension $n$.
A discrete (finitized?) version of this problem is the Finite Field Kakeya conjecture, which was proved by Dvir ...

**13**

votes

**2**answers

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### Number triangle

This question arose just out of curiosity. Note the triangle of 0-1's below, whose construction is as follows. Choose any number, say 53 as done here. The first line of the triangle is the binary ...

**11**

votes

**2**answers

591 views

### Blocking sets in three dimensional finite affine spaces

What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line?
Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = 0,...

**14**

votes

**3**answers

841 views

### Zeta function of Abelian variety over finite field

Let $A/\mathbf{F}_q$ be an Abelian variety of dimension $g$. Suppose one knows $|A(\mathbf{F}_{q^n})|$ for all $1 \leq n \leq g$. Does one know then $\zeta(A,s)$ (equivalently, $|A(\mathbf{F}_{q^n})|$ ...

**9**

votes

**1**answer

725 views

### Related Forms for the Riemann Hypothesis over Function Fields

There are several formulations and consequences of the Riemann Hypothesis over Function Fields (RH, from now on). I am interested in the logical implications between those, and in proofs\references ...

**7**

votes

**2**answers

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### Algorithms to find irreducible polynomials of a given degree

I need to know what are the efficient algorithms to find all the irreducible polynomials of a given degree, say $d$ over a given finite field, say $\mathbb{F}_{p^n}.$
One way is to factorize the ...

**7**

votes

**0**answers

84 views

### Removing rows to reduce the rank

What is the smallest number of rows one can delete from a matrix to reduce its rank (by $1$)? Is there any standard name / notation for this characteristic? Has it been studied?
I am in fact ...

**7**

votes

**0**answers

231 views

### Cyclic shift acting on finite Grassmannian

The (twisted) cyclic shift $(v_1,v_2,\ldots,v_n) \mapsto (v_2,v_3,\ldots,v_n,(-1)^{k-1}v_1)$ acting on the Grassmannian $\mathrm{Gr}(\mathbb{C};k,n)$ of $k$-planes in $\mathbb{C}^n$ is an important ...

**6**

votes

**1**answer

508 views

### Galois action on $p$-adic Tate module of Abelian variety over finite field semisimple?

Let $A,B$ be positive dimensional Abelian varieties over a finite field and $p$ be an arbritrary prime. By Zarhin, Homomorphisms of abelian varieties over finite fields http://www.math.nyu.edu/~...

**6**

votes

**2**answers

2k views

### Calculus over finite fields

$P(x,y,z)$ is a polynomial function on an algebraic surface $S$ in $F_{q}^{3}$. Suppose that the derivative of $P$ along any tangent vector of $S$ is zero. Can we say that $P$ is constant on $S$?
...

**6**

votes

**1**answer

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### How can I solve a cubic equation in a finite field with characteristic 2?

I need to solve the usual cubic equation $x^3 + ax^2 + bx + c = 0$ over a finite field $GF(2^n)$. This is to avoid doing a brute-force Chien search in a BCH decoder.
I read in a paper about an easy ...

**4**

votes

**2**answers

620 views

### 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 ...

**2**

votes

**0**answers

329 views

### Enumerating certain types of permutation polynomials

Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions:
$f(ax) = af(x)$ for all ...

**17**

votes

**0**answers

277 views

+500

### Existence of a polynomial $Q$ of degree $\geq (p-1)/4$ in $\mathbb F_p[x]$ such that $QQ'$ factorizes into distinct linear factors

For all primes up to $p=89$ there exists a product $Q=\prod_{j=1}^d(x-a_j)$ involving $d\geq (p-1)/4$ distinct linear factors $x-a_j$ in $\mathbb F_p[x]$ such that $Q'$ has all its roots in $\mathbb ...

**11**

votes

**2**answers

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### Are $E_n$-operads not formal in characteristic not equal to zero?

This is a short question:
Is it just unproven folklore (yet), or is it definitively known that $E_n$-operads are not formal, if the characteristic of the underlying field is not equal to zero?

**10**

votes

**1**answer

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### How many rich directions does a set in $\mathbb F_p^2$ determine?

$\newcommand{\F}{\mathbb F}$
A subset $P$ of the affine plane $\F_p^2$ is said to determine a direction if there is a line in this direction containing at least two points of $P$.
A set of size $|P|&...

**6**

votes

**0**answers

282 views

### What is known about the $q$-analogue of the simplex?

I am interested in the field with one element. I am thus interested in combinatorial interpretations of the Gaussian binomial coefficients. Richard Stanley's "Enumerative combinatorics" mentions ...

**5**

votes

**1**answer

366 views

### $(n-2)$-blocking sets in $AG(n,2)$

Let's define $k$-blocking set in affine space $AG(n,q)$ a set that meets every coset (translate of subspace) of dimension $k$.
I have seen a lot work related to minimal $(n-1)$-blockings set.
...

**4**

votes

**1**answer

165 views

### $\#E(\mathbb{F}_p) \in \{p,p+2\}$ iff $p$ is of the form $27a^2+27a+7$

Related to this question.
Let $E / \mathbb{F}_p : y^2=x^3+2$.
Numerical evidence up to $2 \cdot 10^5$ suggests:
Conjecture: $\#E(\mathbb{F}_p) \in \{p,p+2\}$ iff $p$ is of the form
$27a^2+27a+7$ ...

**4**

votes

**0**answers

222 views

### Polynomial equations in many variables have solutions (Lang 1952 paper)

I am trying to understand the proof of the following result:
Suppose $F$ is a function field in $k$ variables over an algebraically closed field. Let $f_1,...,f_r \in F[x_1,...,x_n]$ be ...

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votes

**3**answers

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### how to generate the n-torsion group

Hello,
I was curious about the following sentence: "then the $n$-torsion on $E(\overline{K})$ has known structure, as a Cartesian product of two cyclic groups of order $n$" (found at http://en....

**13**

votes

**1**answer

406 views

### Near-linear mappings from $\mathbb F_p$ to $\mathbb R$

$\newcommand{\F}{{\mathbb F}}$
$\newcommand{\R}{{\mathbb R}}$
$\renewcommand{\phi}{\varphi}$
Let $p\ge 5$ be a prime.
If the functions $\phi_1,\phi_2,\phi_3\colon\F_p\to\R$ satisfy $\phi_1(x)+\...

**8**

votes

**0**answers

293 views

### A strong sum-product "for translates" in finite fields

In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...

**6**

votes

**1**answer

243 views

### Which criteria for "uniformly splitting" polynomials?

Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials ...

**5**

votes

**1**answer

276 views

### 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,...

**5**

votes

**1**answer

767 views

### Full-rank rectangular matrices over GF(2)

Given positive integers $k$, $m$, $n$, let $A$ be an $m \times n$ matrix over $GF(2)$ constructed as follows. Let $X_1, \ldots, X_m$ be independent random subsets of $\{1,\ldots,n\}$ with cardinality ...

**4**

votes

**1**answer

258 views

### Large gaps in Singer planar difference sets?

By a classical result of Singer (1938), for a prime number $p$ the cyclic group $C_n$ of order $n=1+p+p^2$ contains a subset $D$ of cardinality $|D|=1+p$ such that $DD^{-1}=C_n$. Such set $D$ is ...

**3**

votes

**2**answers

215 views

### Largest $A\subset \mathbb{F}_2^n$ such that no two $a\neq b$ in $A$ add to an element of $A.$

If such a set $A$ of size $m$ exists, all its admissible pairwise sums must lie in its complement, thus
$$
\binom{m}{2} \leq 2^n-m,
$$
which gives $$m\leq 2^{(n+1)/2}\qquad (1)$$.
Edit: The upper ...

**1**

vote

**1**answer

832 views

### maximal number of mutually orthogonal vectors

Let $F$ be a field, $n$ be a positive integer. Denote by $h_{F}(n)$ the maximal dimension of a subspace $X\subset F^n$ such that $(x,y)=0$ for any two (not necessary distinct) vectors $x,y\in F^n$, ...

**0**

votes

**1**answer

1k views

### 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 ...

**6**

votes

**1**answer

292 views

### 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} | \...

**6**

votes

**3**answers

386 views

### On the rank of a matrix $S$ with coefficients in $\mathbb F_{2^m}$

Let $\mathbb F$ be a finite field with characteristic 2 and let $S \in M(2k, 2k, \mathbb F)$ be the matrix defined as follows
$$ S=\left[\begin{array}{ccccccc}
0 & \...

**6**

votes

**0**answers

527 views

### 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 ...

**5**

votes

**1**answer

426 views

### More expanders?

Having received several exhausting answers to my recent question about
the expansion properties of a certain graph, I now wonder whether anything is
known on the following graphs of a similar nature:
...