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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Is there an available English translation for Artin's "Quadratische Körper im Gebiete der höheren Kongruenzen"?

Otherwise, is it reasonable to work through the German edition with only a basic knowledge of German?
delpsi's user avatar
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History of algebraic geometry over finite fields

My question is of historical nature: when did mathematicians start studying algebraic geometry over finite fields in a systematic way, and who were the main driving forces ? Did it start with Weil (...
THC's user avatar
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Does the Plotkin bound mean that one can not achieve the Singleton bound asymptotically?

I am a little confused with the relationship between various bounds for error correcting codes. Does the Plotkin bound mean that one can not achieve the Singleton bound asymptotically? That is, is ...
liu_c_6's user avatar
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Moduli space of abelian surfaces

Let $K$ be a finite field with algebraic closure $\overline{K}$. The $j$-invariant gives a bijection between the the affine line $\mathbb{A}_K^1$ and $\overline{K}$-isomorphism classes of elliptic ...
Sebastian Monnet's user avatar
6 votes
1 answer
304 views

Number of points on a linear algebraic group over a finite field

Let $G$ be a linear algebraic group defined over a finite field $\mathbb{F}_q$ as a variety of dimension $d$. What would be a good, simple lower bound for $G(F_q)$? One can get something fairly nice ...
H A Helfgott's user avatar
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Why all supersingular elliptic curves over $\bar{\mathbb{F}_p}$ are isogenous?

Lemma 3.2.1 in Baker, González-Jiménez, González, Poonen, "Finiteness theorems for modular curves of genus at least 2", Amer. J. Math. 127 (2005), 1325–1387. enter image description here I ...
HaomengXu's user avatar
4 votes
2 answers
497 views

On a certain equation in finite fields

I am interested in the following question. Let $q$ be a prime power and let $\mathbb{F}_q$ be the finite field of cardinality $q$. Suppose $q>61$. Is it true that, for every $b\in \mathbb{F}_q$ and ...
Pablo Spiga's user avatar
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A characterisation of full subgroups of $\mathrm{GL}_n(\mathbf{F}_p)$

Let $p\geq 5$ be a prime and $\mathbf{F}_p$ a finite field of characteristic $p$. A subgroup of ${\rm GL}_n(\mathbf{F}_p)$ is full if it contains ${\rm{SL}}_n(\mathbf{F}_p)$. When $n=2$, we have the ...
stupid boy's user avatar
1 vote
1 answer
428 views

Artin's conjecture for polynomials and rational functions over finite fields

Artin's conjecture on primitive roots over the integers states that a given integer $0\ne h\in \mathbb{Z}$ that is neither a square number nor $-1$ is a primitive root modulo infinitely many primes $p$...
user500926's user avatar
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Number of full-rank binary matrices with given column Hamming weights [closed]

What is the number of $m \times n$ matrices over $\mbox{GF}(2)$ that share the following constraints : They have full rank ($\mbox{rank} = m$, given that $m<n$). Their columns have the given ...
Sapiens's user avatar
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Easy bound on number of points on a variety over a finite field

Let $V$ be a closed algebraic set in $\mathbb{A}^n$ defined over a finite field $K$, with irreducible components $Z_1,\dotsc,Z_m$. Let $D = \sum_i \deg Z_i$ and $d = \max_i \dim Z_i$. Then $$|V(K)|\...
H A Helfgott's user avatar
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3 votes
1 answer
281 views

What do we know about the $\mathbb{F}_{q^2}$-curve $X^{q-1} + Y^{q-1} + Z^{q-1}$?

People have thoroughly studied the Hermitian $\mathbb{F}_{q^2}$-curve $H: X^{q+1} + Y^{q+1} + Z^{q+1} = 0$. What do we know about the similar $\mathbb{F}_{q^2}$-curve $C: X^{q-1} + Y^{q-1} + Z^{q-1} = ...
Dimitri Koshelev's user avatar
2 votes
1 answer
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Inner product over finite field

sorry for informals but is my first post. In Coding theory (exactly in Coding theory a first course - San ling and Chaoping) found this definition: $\langle , \rangle :\mathbb{F}_q^n\times\mathbb{F}_q^...
Javier Astorga's user avatar
3 votes
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274 views

Approximate versions of Segre's Theorem

Consider projective $2$-space over a finite field of odd prime characteristic $p$. We say a set of points, $A$, in this space is an arc if any line meets it in at most two points. We say that an arc ...
Mark Lewko's user avatar
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How many ways are there to pick $n$ points on the finite affine plane $(\Bbb F_q)^2$ such that no three are collinear?

How many ways are there to pick $n$ points on the finite affine plane $(\Bbb F_q)^2$ such that no three are collinear? For example, how many ways can we pick $5$ points on $\Bbb F_{32}\times\Bbb F_{32}...
Akiva Weinberger's user avatar
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Will an integer program to deterministically factor integers help derandomize $\mathbb F_q[x]$ factoring?

There are many analogies between the objects $\mathbb F_q[x]$ and $\mathbb Z$. Supposing there is a fixed (say $10^9$) dimension linear integer program (describable without any objective function) in ...
Turbo's user avatar
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2 votes
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A generalisation of Moore-Ore criterion?

Let $K$ be a field of characteristic $p>0$ and $q=p^f$. One supposes that $\mathbb F_q$ is embedded in $K$. One considers elements of $K$: $w_1,\dotsc,w_s$. Assume there exists $b_1,\dotsc,b_s$ in $...
joaopa's user avatar
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Solving efficiently a quadratic equation in a large finite field of characteristic two

I'm trying to solve efficiently a quadratic equation in the finite field $\text{GF}(2^{128})$ represented as $(\mathbb{Z}/2\mathbb{Z})[x] / (x^{128} + x^7 + x^2 + x + 1)$. Until now, I came across ...
ocalex86's user avatar
3 votes
0 answers
87 views

Equirepartition of sums for large multisets in subsets of finite fields

Let $p$ be a prime number and let $\mathcal A$ be a subset of $a\leq p$ distinct elements in $\mathbb F_p$. We denote by $\mathcal M_k(\mathcal A)$ the set of all ${k+a-1\choose k}$ multisets ...
Roland Bacher's user avatar
1 vote
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250 views

Reductions of a system of equations at various primes

Let $f_1, \dots, f_n$ be a finite set of polynomials in the polynomial ring $Z[x_1, \dots, x_m]$. At a prime $p$, let $N_p$ be the number of solutions $x=(x_1, \dots, x_m)\in (\mathbb{Z}/p\mathbb{Z})^...
Anwesh Ray's user avatar
4 votes
1 answer
218 views

Intersection of the general linear group $\text{GL}_n(\mathbb{F}_q)$ and an affine subspace over a finite field

Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\text{M}_n(\mathbb{F}_q)$ be the vector space of $n \times n$ matrices over $\mathbb{F}_q$, let $\text{GL}_n(\mathbb{F}_q)$ be the group of $...
Kleo's user avatar
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18 votes
2 answers
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Has anyone seen this construction of the Weil representation of $\mathrm{Sp}_{2k}(\mathbb{F}_p)$?

$\def\FF{\mathbb{F}}\def\CC{\mathbb{C}}\def\QQ{\mathbb{Q}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}\def\PGL{\text{PGL}}$Let $p$ be an odd prime. The Weil representation is a $p^k$-...
David E Speyer's user avatar
6 votes
0 answers
240 views

Cardinality of a polynomial image $\pmod{p^n}$

Given $P(x) \in \mathbb{Z}[x]$ a polynomial, $n$ a positive integer and $p$ a prime, there is a result that relates $|\text{Im } P \pmod{p^n}|$ with $|\text{Im } P \pmod{p^{n+1}}|$ perhaps in terms of ...
Rfluid's user avatar
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Representing an $m$ dimensional quadratic polynomial as a polynomial on $\mathbb F_{q^m}$

We can represent $\mathbb{F}_{q^m}$ as $\mathbb{F}_q[\alpha]$ where $\alpha$ is root of an irreducible $m$-degree polynomial on $\mathbb{F}_q$. By sending $\sum_{i=0}^{m-1} c_i\alpha^i \mapsto (c_{m-1}...
user's user avatar
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5 votes
1 answer
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How to make Burnside's formula compatible with point counting for varieties over finite fields?

If $G$ is a finite group acting on a finite set $X$, we have Burnside's formula that counts the number of orbits $|X/G|$ as: $$ |X/G| = \frac1{|G|} \sum_{g\in G} |X^g|, $$ with $X^g$ being the set of ...
bernardorim's user avatar
0 votes
1 answer
231 views

Product of subspace and its inverse

$\DeclareMathOperator\GF{GF}$Let $R=\GF(q)$ be a finite field with $q=p^r$ elements, where $p$ is a prime number, $S=\GF(q^n)$ be an extension of $R$, where $n\in \mathbb{N}$, $n\geq 2$ and let $K=\GF(...
Mikhail Goltvanitsa's user avatar
7 votes
2 answers
391 views

Sum of two $n$th powers in finite fields

Let $q$ be a prime power, let $n$ be a positive integer and let $\mathbb{F}_q$ be the finite field of cardinality $q$. I have some computational evidence that the set $$\{x^n+(-1)^nay^n:x,y\in\mathbb{...
Pablo Spiga's user avatar
0 votes
0 answers
167 views

How many elements have a "small" order in a finite field?

I'm hoping that this is an easy question for someone. How many elements can we expect to have multiplicative order at most $n^{1/c}$ in one of the finite fields $\mathbb{F}_p$ with $p$ prime with $n \...
Matt Groff's user avatar
4 votes
0 answers
125 views

Statistics about existence of rational points on a curve over $\mathbb{F}_q$

I wish to ask the naive question: if we write down a random curve $C$ over $\mathbb{F}_q$, what can be said about the probability that $C(\mathbb{F}_q)=\emptyset$? Of course, this depends on the ...
Cheng-Chiang Tsai's user avatar
6 votes
0 answers
255 views

Number of square-free polynomials over a finite field - a combinatorial interpretation?

Cross-posted from MSE. The question has remained unanswered for six years but I still like it! One can show using zeta functions that the number of (monic)square-free polynomials of degree $n$ over a ...
Asvin's user avatar
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1 vote
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80 views

Constructing k-wise independent variables over a general set

We have seen in class a polynomials based construction that builds in $O(n^k)$ time, $n$ random variables, $k$-wise independent, over a field with $n$ elements. More specifically, you generate all the ...
tamir's user avatar
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1 vote
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109 views

Maximal subsets of affine or projective space with no three collinear points

Let ${\mathbb A}^n_q$ and ${\mathbb P}^n_q$ be affine and projective spaces of dimension $n$ over a field of order $q$. Say that a subset of either ${\mathbb A}^n_q$ or ${\mathbb P}^n_q$ is generic if ...
Steven Landsburg's user avatar
0 votes
1 answer
86 views

Enumerating (i.e. generating one by one) matrices of given rank over a finite field

Let be given positive integers $m,n,r$, with $r \leq \min(m, n)$, and a finite field of $q$ elements $\mathbb{F}_q$. I'm looking for an efficient algorithm to enumerate (i.e., generate one by one) all ...
Kleo's user avatar
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4 votes
0 answers
101 views

Visiting zero-sum triples in a vector space

Is it true that for any set $A\subset\mathbb F_5^n$ satisfying $A\cap(-A)=\varnothing$, there is a subset $A'\subseteq A$ such any triple $(a,b,c)\in A\times A\times A$ with $a+b+c=0$ has exactly one ...
Seva's user avatar
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3 votes
1 answer
169 views

Largest subset of quadratic residues with no pair of elements differing by 1

In $\mathbb{F_p}$, $p$ prime what is the larget subset $S$ of quadratic residues with no pair of elements differing by 1? In this related question Seva gives an example: "...assuming $p\equiv\...
Ivan Meir's user avatar
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4 votes
0 answers
127 views

7D simple Lie algebras over $\mathbb{F}_3$

Up to isomorphism, what are all the seven-dimensional simple Lie algebras over the field with three elements?
Daniel Sebald's user avatar
11 votes
2 answers
2k views

Algebraic geometry over the complex numbers, and beyond

My question basically is very simple: when did mathematicians start to do algebraic geometry "outside the complex numbers" ? In the old days, algebraic geometry was solely done over the ...
THC's user avatar
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8 votes
1 answer
329 views

The distribution of certain Galois groups

Let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Let $G_p^+$ be the Galois group of the polynomial $f(x)-y$ over $\overline{\mathbb{F}}_p(y)$ and $G_p$ be the Galois group of the ...
Alexander Kalmynin's user avatar
2 votes
2 answers
219 views

Sum over exponentiated bilinear form in finite-field vector space

Let $A$ be a linear map over the finite-field vector space $(\mathbb F_2)^n$, i.e., an $\mathbb F_2$-valued $n\times n$ matrix, not necessarily symmetric. I'm interested in the sum $$Z(A) = \sum_{X\...
Andi Bauer's user avatar
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11 votes
1 answer
1k views

Pointless groups

This question now has two sequels, Pointless groups II (to which @R.vanDobbendeBruyn gave a counterexample for an infinite, imperfect field) and Pointless groups III, both using revised wording ...
LSpice's user avatar
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3 votes
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Is there a bijection between elements in algebraic closure of F2 and all bi-infinite periodic sequences made of 0 and 1, filling the properties below?

(And if so, how can I describe the "multiplication" on the sequence?) We consider a bijection, denoted by $f$, from the algebraic closure of $\mathbb{F}_2$, named $\bar{\mathbb{F}_2}$, to ...
Ginger's user avatar
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1 vote
0 answers
141 views

Counting non-zero Gramians of Grassmanians over finite field

In case of $\mathbb{F}_{2}$, we can obtain the number of all reduced row echelon forms (so called Grassmannians) for some m$\times$n full rank matrices by the following gaussian polynomial, $$ \binom{...
mathcat's user avatar
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3 votes
0 answers
93 views

Are supersingular K3 surfaces unirational?

There is a conjecture due to Artin, Rudakov, Shafarevich, Shioda that supersingular K3 surfaces over a finite field are unirational. This paper claims to prove this result but it has had a recent ...
Asvin's user avatar
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0 votes
0 answers
96 views

Invertible matrices with bounded nonnegative coefficients

I am teaching a class in linear algebra and I asked myself the following question: what is the chance to get an invertible matrix if I write a random one? My impulsive answer is "very likely"...
Andrea Marino's user avatar
1 vote
0 answers
303 views

How to show the minimal polynomial of primitive n-th root of unity on prime field with characteristic p is the following? [closed]

How to show : w, the primitive n-th root of unity over prime field F with characteristic p, gcd(n,p)=1, is $(x-w)(x-w^p)(x-w^{p^2})...(x-w^{p^{r-1}})$ where r is the smallest positive integer s.t. $p^...
mathfan's user avatar
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2 votes
4 answers
636 views

How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?

How to prove without using advanced theorems that quaternions algebra $H = \genfrac(){}{}{-1,-1}{\mathbb{Z}_p}$, where $p$ is prime that $H \cong\operatorname{Mat}_2({\mathbb{Z}_p})$? My ideas: I ...
Joseph Jordan's user avatar
1 vote
1 answer
175 views

Known estimate for gaussian sum $\sum_{x \in \mathbb{F}_q} \psi( a x^m + b x^n)$?

Let $\mathbb{F}_q$ be a finite field, $\psi$ be a non-trivial additive character over $\mathbb{F}_q$, and $a, b \in \mathbb{F}_q$ constants. Is there any known estimate for the gaussian sum $$\sum_{x \...
José's user avatar
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1 vote
1 answer
282 views

A "semi-genetic" definition of addition and multiplication in the field $\operatorname{On}_p$?

Let $+,\cdot$ denote multiplication in $\mathbb{N}_0$. The addition and multiplication in $\operatorname{On}_p$ are denoted $\oplus, \otimes$. Recursive definition of addition: $$x \oplus y := ((x+y) \...
mathoverflowUser's user avatar
3 votes
0 answers
124 views

Order of elements $\gamma$ and $1-\gamma$ in $\mathbb{F}_q$

I'd like to find if possible the orders possible for $1-\gamma$ given a $\gamma\in\mathbb{F}_q$ of given order $\mathcal{o}$ (where $q=p^f$). Which is clear is that these order have the same degree (...
Gabriel Soranzo's user avatar
1 vote
1 answer
180 views

Order of roots for a polynomial $P\in\mathbb{F}_p[T]$

Let $P\in\mathbb{F}_p[T]$ (not supposed irreducible). All roots $\xi$ of $P$ have a certain order $k$ such that $\xi^k=1$. Question: is it possible to know the order of the roots of the given ...
Gabriel Soranzo's user avatar

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