# 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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### 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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100 views

### 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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**1**answer

164 views

### 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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94 views

### On the sum $\sum_{x=0}^{(p-1)/2}(\frac{x^{4n}+cx^{2n}+d}p) $ with $p$ an odd prime

Let $p$ be an odd prime, and let $n$ be a positive integer. For $c,d\in\mathbb Z$ we define
$$F_p^{(n)}(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^{4n}+cx^{2n}+d}p\right),$$
where $(\frac{\cdot}p)$ is ...

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79 views

### 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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85 views

### Is there a notion of linear $\mathbb F_q[x]$ program?

A feasibility linear integer program is essentially deciding if a convex polytope represented by a set of inequalities has an integer point and a minimization linear integer program is to find the ...

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**1**answer

246 views

### 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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**2**answers

198 views

### 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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203 views

### 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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140 views

### 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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**2**answers

286 views

### 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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35 views

### Computing with a vector subspace equipped with a prescribed basis over finite fields

Let $U$ be a subspace of the finite dimensional vector space $V$ over a field $\mathbb{k}$. Let $B_V$ and $B_U$ be fixed bases for $V$ and $U$ respectively. Let $u \in U$ and let's give ourselves $[u]...

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**1**answer

269 views

### 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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124 views

### 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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85 views

### 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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94 views

### $\mathbb{F}_q$-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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**1**answer

222 views

### 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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**1**answer

236 views

### 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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**1**answer

189 views

### 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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**2**answers

337 views

### 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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**1**answer

141 views

### 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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**1**answer

101 views

### 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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163 views

### 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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249 views

### 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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186 views

### 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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72 views

### 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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239 views

### 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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76 views

### 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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**1**answer

61 views

### 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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142 views

### 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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**1**answer

129 views

### 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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66 views

### 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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**1**answer

161 views

### 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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31 views

### 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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**1**answer

165 views

### 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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121 views

### 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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207 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 ...

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170 views

### 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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**1**answer

392 views

### 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 $F$ ...

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**1**answer

368 views

### 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|&...

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**1**answer

139 views

### Is the Adjoint Action self dual over finite fields?

Given a finite group $G$, and representation $\rho: G \to H(\mathbb{F}_q)$ where $H$ is some classical algebraic group ($Gl$, $Sl$, $O$, $SO$, $SP$, $GSP$, $U$, etc), is the induced Adjoint ...

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78 views

### Trace of Symmetric matrices in fixed rank

I am solving some problem related to symmetric matrices over a finite field $\mathbb{F}_q$ and I am stuck at the following problem:
For every $a\in\mathbb{F}_q $, let $S_a(t,m)$ be the set of all $m\...

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243 views

### How many conjugacy classes of elementary abelian subgroups of rank $2$ does $GL_{n}(Z / pZ)$ have?

Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z}
)$. Consider the set $U$ of upper-triangular matrices of $G$
having entries of $1$ on the diagonal. The cardinality of $U$ is $p^{\...

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250 views

### Do we expect abelian varieties (and “Artin motives”) to generate the Grothendieck ring of varieties over a finite field?

The Tate conjecture implies that the category of motives over a finite field is generated (as tensor category) by the motives of abelian varieties and Artin motives. See [1] for details.
Let $K(\...

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**1**answer

47 views

### An Extension of an $\operatorname{MDS}$ Code over $\operatorname{GF}(2^q)$

Let $q$ be a power of $2$.
Assume that elements of the finite field $\operatorname{GF}(q)$ are denoted by $\beta_i$
for $0\leq i \leq q-1$. We divide elements of $\operatorname{GF}(q)$ in two parts ...

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**4**answers

382 views

### Show that sets are equal

Let $X=\{x_1,x_2,...,x_n\}$ and $Y=\{y_1,y_2,...,y_n\}$ be sets over a finite field $F$ with $p=char(F)>2$. Assume
$$x_1^k+x_2^k+...+x_n^k=y_1^k+y_2^k+...+y_n^k,\ 1\leq k\leq n$$
I wanna ...

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**0**answers

106 views

### Roots of polynomials over Z/p^kZ

Over a finite field, such as Z/pZ, the number of roots of a polynomial is no larger than the degree. I'm interested in how does this generalize to Z/p^kZ.
I'm sure that this has been looked at, but I ...

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**1**answer

233 views

### Herbrand-Ribet and Mazur-Wiles for function fields

Is there a version of Herbrand-Ribet or Mazur-Wiles (relating divisibility of class groups to special values of L-functions) for functions fields (over finite fields)?
Probably the proofs would have ...

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vote

**1**answer

82 views

### Generalization of a field norm to the linear group

I've been struggling with this question for some days now. Let $K$ be a field extension of $k$, and $x$ an invertible linear transformation of the $K$-vector space $V$. If we consider $V$ as a $k$-...

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**1**answer

190 views

### Supersingularity and Roots of Unity of Zeta Functions

Let $C$ be smooth projective curve defined over a finite field $\mathbb{F}_q$. Let $$Z(C,u)=\exp(\sum_{n \ge 1} N_r(C) u^r/r) \in \mathbb{Z}[[u]]$$ be its zeta function, where $N_r$ is the number of $\...