# 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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### The number of unitary circulant matrices over a finite field $\mathbb{F}_{q^2}$

I asked this question in MSE few days ago but there was no response.
Suppose $\mathbb{F}=\mathbb{F}_{q^2}$, where $q$ is a prime power. The conjugate of elements in $\mathbb{F}$ is defined by $\...

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### Is an integral fusion category pseudo-unitary over a finite field?

Here are two propositions in the book Tensor Categories:
Proposition 9.5.1. A pseudo-unitary fusion category admits a unique
spherical structure.
Proposition 9.6.5. Let $\mathcal{C}$ be ...

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### Reducible polynomial among sequence of polynomials

Let $a_1$ and $a_2$ be two elements of a finte field $\mathbb{F}_{2^m}$ of even characteristic and $a_1^2\neq a_2$. Is it true that there always exists an element $a\in\{a_1,a_2,a_3,\ldots,a_{2^m}|a_{...

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### Number of elements in $\mathrm{GL}(n,p)$ with maximal order

I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$.
I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...

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### Counting the image of a map of varieties using the trace formula

Suppose $f: X\to Y$ is a finite map of varieties over a finite field $\mathbb F_q$. Is there an etale constructible $\mathbb Q_\ell$ sheaf $\mathscr F$ on $Y$ which counts the number of rational ...

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### Probability that random points are affine subspace

I asked this question in Math stack exchange, but I think it is more relevant here.
Suppose that $\mathbb{F}_q$ is a finite field with $q$ elements. Let $U = \{u_1, \ldots,u_m\}$ be a set of $m$ ...

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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 ...

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### Affine projection of polynomials for a given set of points

(Not sure this question fits here, I will remove it in case it doesn't)
Let $F_{\mathrm{ML}}[x_1, \ldots,x_n]$ denote the set of multilinear polynomials over a finite field $F=\mathbb{F}_q$ (i.e. a ...

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### Does “tensoring” with a fixed field preserve Galois extensions of finite fields?

Let $K$ be a (possibly infinite) field of characteristic $p$, and $L$ be a finite field extension of $\mathbb{F}_p$, so that $L$ is finite and $L/\mathbb{F}_p$ is Galois. Suppose $K \otimes_{\mathbb{F}...

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### zeros of a polynomial in three variables over finite field

Consider a homogeneous symmetric polynomial $f(x,y,z)=(x+y+z)((xy)^{q-1}+(yz)^{q-1}+(zx)^{q-1})+x^{2q-1}+y^{2q-1}+z^{2q-1}$ of degree $2q-1$, where $q$ is an odd prime power.
Under what condition ...

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### For which (g,q) does there exist a supersingular curve?

We say a curve over a finite field $\mathbb F_q$ is supersingular if it's Frobenius eigenvalues (on $H^1(X,\mathbb Z_\ell)$) are of the form $q^{1/2}\alpha$ for $\alpha$ a root of unity.
As far as I ...

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### Reference / Survey for finite field analog number theory

It is folklore that many number theoretic results on prime numbers have a simpler-to-prove finite field analog. For example, on the one hand, the proof of the Prime Number Theorem
$$\#\{\text{prime ...

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### Upper bound for an exponential sum involving characters of a finite field

Let $q = p^n $ be a prime power, $\alpha\in\mathbb{F}_{q} $
a primitive element of the finite field $\mathbb{F}_q$ and denote by $\chi $ a non-trivial additive character of $\mathbb{F}_{q} $. Set
$\...

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### describing embedding $U_3(q)<O_6^-(q)$, $q$ even

Let $q=2^k$. I need to explicitly construct $U_3(q)$ as a subgroup of $G=GO_6^-(q)$. It is well-known that
$G\cong U_4(q)$, and as a subgroup of the latter one has $U_3(q)$ fixing a non-isotropic ...

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138 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,...

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### A Lefschetz style formula for the $\ell^\infty$ torsion of an Abelian variety over a finite field

Let $A/\mathbb F_q$ be an abelian variety over a finite field. Define $A_\ell = A[\ell^\infty](\mathbb F_q)$, the $\ell^n$ ($n\geq 0)$ torsion points defined over the base field. I can assume $\ell \...

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### Partitioning a given set into root set of two polynomials or non zero set of two polynomials simultaneously

Let $p,q$ be two prime numbers and let $p<q.$ Also let, $\mathbb{Z}_p$ and $\mathbb{Z}_q$ denote the fields formed by integers modulo $p$ and modulo $q$ respectively (with respect to the modulo $p$ ...

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### Inseparable field extensions of degree p and linear independence

Let $F$ be a field of characteristic $p$; let $\alpha \in F$ such that $\alpha \neq \beta^p$ for any $\beta \in F$, and let $K := F(x)$ where $x=\sqrt[p]{\alpha}$.
Is it true that the elements $1,(x-...

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### Does the cardinality of coordinate projections of the rational points of affine varieties over finite fields also tend to $\infty$?

We know (basically by Lang-Weil) that for an absolutely irreducible n-dimensional affine variety $V$ the cardinality $\#V(F_{l})$ tends to $\infty$ for $l$ large enough. We could now look at the set ...

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### General linear group action on extensions of finite fields

Let $q$ be a prime power. Let $\mathbb{F}_q$ be the finite field with $q$ elements. Then $\mathbb{F}_{q^n}$ is a field extension of $\mathbb{F}_q$ of degree $n$ and can be considered as an $n$-...

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### Why should it be hard to generalize Dvir's proof of the finite field Kakeya conjecture to the Euclidean case?

Let $q$ be prime and let
$q\delta \sim 1.$ Let $K$ be any set of $Cn\delta$-separated tubes in $B(0,2)$, where $C_n$ is some constant depending on $n$. Let us consider a grid of $q^n$ points scaled ...

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### Generalized Moore matrix

Let $a_1,a_2,\cdots,a_m$ be element of a finite field $\mathbf{F}_{q^n}$ of order $q^n$, where $q$ is a prime power. It is well known that the matrix:
$$
\begin{array}{cccc}
a_1 & a_1^{q}...

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### Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$

For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial ...

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### Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials

Let $n>1$ and $p$ be an odd prime with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. Notice that, since $p-1 \mid n-1$, we have $T^p - T \mid T^n-T$ in $\mathbb{F}_p[T]...

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### Density of rational points over finite fields, an estimate of Lang-Weil constant

Let $X\hookrightarrow\mathbb P^n_{\mathbb F_q}$ be a geometrically integral hypersurface over the finite field $\mathbb F_q$ of degree $\delta$. In order to estimate the number of its rational points, ...

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### Is there any way to solve this equation without knowing the inverse modulo? [closed]

Suppose I have an arbitrary 256 bit number $m$ another secret number $k$ of the same bit length, and then I multiply them both modulo a 256 bit prime number $p$ to get $c$ as follows:
$$
c = (m\cdot k)...

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### What's the meaning of the nontrivial zeros of Selberg zeta function?

In the case of arithmetic variety over finite field, the zero points of the Hasse-Weil zeta function reflect the pure weights (i.e. dimension). On the other hand, in the case of the Selberg zeta ...

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### Cubic polynomials over finite fields whose roots are quadratic residues or non-residues

For a cubic polynomial $f(x)=x^3+x^2+\frac{1}{4}x+c$ over $\mathbb{F}_q$, where $q$ is a odd prime power, I find that for a lot of $q$, there does not exist $c\in\mathbb{F}_q$ such that $f$ has three ...

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### Bivariate polynomial divisibility test of Spielman

Setup
In his thesis (lemma 4.2.18, p. 97-98) Spielman describes a divisibility test for bivariate polynomials $E,P\in k[X,Y]$, where $k$ is a field (of positive characteristic for what I'm interested ...

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### What fraction of multivariate polynomials with bounded individual degrees are irreducible?

How many polynomials in $\mathbb{F}_p[x_1, \dots, x_m]$ with degree at most $d-1$ in each variable $x_i$ are irreducible?
Here $m$ and $d$ are positive integers, $p$ is a prime, and $\mathbb{F}_p$ ...

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### On sparse $0/1$ linear equations solvable with compressed sensing

If you have a system of $m$ linearly independent equations in $n$ variables with domain $0/1$ and we know there is at least one solution with at most $d$ variables to be $1$ then if $m$ at least a ...

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### How many matrices $C \in \mathrm{M}_3(\mathbb{F}_q)$ such that $\det(C-A)=\det(C-B) = 1$?

I am studying the special unit-graph $G$ on $M_3(\mathbb{F}_q).$ Now, I want to estimates the upper bound for the second largest eigenvalue of adjacency matric of $G.$ One of questions that I need is ...

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### Given a symmetric polynomial in F_q, write it in terms as elementary symmetric polynomials. How to find out the coefficient?

Consider the finite field $F_q$, where $q$ is a power of an odd prime and $N$ is a power of $q$. We have a homogeneous symmetric polynomial
$$
E_{l,s}(x) = \sum_{\substack{l_1+l_2+\cdots +l_s=l \\ l_i\...

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### Finding the minimum dimension of $\operatorname{SL}_n(\mathbb{F}_q)$'s nontrivial real representations

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and $\operatorname{SL}_n(\mathbb{F}_q)$ the special linear group in $n$ variables.
What is the minimum dimension of nontrivial real ...

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### Variation of Gauss/Jacobi sums on a variety

Let $V \subset \mathbb P^n$ be a nice (smooth, projective?) variety over a finite field $\mathbb F_q$. Let $\chi_0,\chi_1,\dots,\chi_{n}: \mathbb F_q^\times \to \mathbb Q(\mu_{q-1})$ be multiplicative ...

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### Character values of principal series representations of $GL_n(\mathbb{F}_q)$

Let $P_{\alpha}$ be the principal series representation of $GL_n(\mathbb{F}_q)$, where $\alpha = ( \alpha_1, \alpha_2, \cdots, \alpha_n)$ and $\alpha_i : \mathbb{F}_q^* \rightarrow \mathbb{C}^*$.
...

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### Étale fundamental group of multiplicative group over an algebraically/separably closed field

This is a repost of my question here.
Do we know the structure of the étale fundamental group $\pi^\text{et}_1(\mathbb{G}_{m,K^\text{sep}})$ of the multiplicative group, for a given field $K$? For ...

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### How to prove $(\phi-1)(\phi-2)…(\phi-p) = \sqrt{5} + p\left(\frac{1}{2}+A\sqrt{5}\right) \bmod p^2$?

We consider the solution of $x^2=x+1$ and denote them as $\phi=\frac{1}{2}(1-\sqrt{5}),\bar\phi=\frac{1}{2}(1+\sqrt{5})$.
Suppose $\phi \not\in \mathbb{F}_p$. In other words, $\sqrt{5} \not \in \...

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### Which fields and schemes “have enough finite residue fields”?

I am looking for assumptions on the spectrum $S$ of a field $K$ that ensure the following: there exists an excellent noetherian finite dimensional (integral) scheme $S'$ such that $S$ is its generic ...

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### A functor on Abelian varieties corresponding to this operation on Weil numbers

Let $A/\mathbb F_q$ be an abelian variety over a finite field with Weil numbers $q^{1/2}\alpha_1,\dots,q^{1/2}\alpha_n$.
Consider the numbers $q^{d/2}\alpha_1,\dots,q^{d/2}\alpha_n$. These are still ...

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### The product of two supersingular elliptic curves is independent of which ones we pick

In a comment on this MO question, Qing Liu says "In positive characteristic p, if you take two supersingular elliptic curves $E_1,E_2$, then $E_i×E_j$ is isomorphic to $E^2_1$ for any pair $i,j$."
...

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### Counting special metrics on finite fields

Define a Galois coding norm of degree n as a map $|\space| : \Bbb F_{2^n}\rightarrow {\Bbb Z}$ with the following properties :
(I) $(\Bbb F_{2^n},|\space|)$ is a self-orthogonal code ; i.e. $(x,y)\...

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203 views

### Best strategy for a combinatorial game

Consider the following scenario. We have 20 balls and 100 boxes. We put all 20 balls into the boxes, and each box can contain at most one ball.
Now suppose we are given 5 chances to pick 20 out of ...

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63 views

### p-groups embedded into Sylow subgroups

Let $p$ be a prime number, $q$ a power of $p$ and $P$ be a finite $p$-group. $P$ is isomorphic to a subroup of p-Sylow subgroup of
the symmetric group $S_{\mid P\mid}$ (Theorem of Cayley)
the ...

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107 views

### Under what conditions on $A$ and $v$ is the size of the sumset $v \cdot A + A$ over $\mathbb{F}_p$ equal or close to $|A|^2$?

Let $p$ be a prime, let $A$ be a subset of $\mathbb{F}_p$, and let $v \in \mathbb{F}_p \setminus \{0\}$.
Under what conditions is $|v \cdot A + A|$ (that is, $|\{ va + b : a \in A,\ b \in A \}|$) ...

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### Sums of squares in fields

Which fields $k$ have the property that any sum of squares is a square ?
Are there elegant characterizations and/or classifications known for this type of field ?
And what if we replace "fields" by "...

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### 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 ...

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431 views

### (Barely) linearly independent vectors over $\mathbb{Z}/2\mathbb{Z}$

Let $V$ be a vector space over $\mathbb{Z}/2\mathbb{Z}$. Can there be a set $S$ of $2 n$ vectors in $V$ such that any $n$ vectors in $S$ span a space of dimension exactly $n-1$, but no $n$ vectors $...

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167 views

### Elliptic curve over Galois Field, Blockchain [closed]

I am interested in the elliptic curve
$$
y^2 = x^3 + 7
$$
where both $x$ and $y$ are in the finite residue class field $F_p$ with $p=2^{256}-2^{32}-2^9-2^8-2^7 -2^6-2^4 -1$. Those parameters are used ...

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### Irreducibility of a family of hypersurfaces over $\mathbb{F}_p$

Let $Q \in \mathbb{F}_p[x,y,z]$ be a geometrically irreducible polynomial. If it helps, suppose $Q$ is a quadratic form. Consider the parameterization
$$x = u(r) , \ \ y = v(s), \ \ z = w(t),$$
where ...