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?
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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 (...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $p$...
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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 ...
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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)|\...
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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} = ...
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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^...
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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 ...
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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}...
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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 ...
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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 $...
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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 ...
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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 ...
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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})^...
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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 $...
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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$-...
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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 ...
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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}...
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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|,
$$
with $X^g$ being the set of ...
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Product of subspace and its inverse
$\DeclareMathOperator\GF{GF}$Let $R=\GF(q)$ be a finite field with $q=p^r$ elements, where $p$ is a prime number, $S=\GF(q^n)$ be an extension of $R$, where $n\in \mathbb{N}$, $n\geq 2$ and let $K=\GF(...
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Sum of two $n$th powers in finite fields
Let $q$ be a prime power, let $n$ be a positive integer and let $\mathbb{F}_q$ be the finite field of cardinality $q$. I have some computational evidence that the set $$\{x^n+(-1)^nay^n:x,y\in\mathbb{...
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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?
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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 ...
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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 ...
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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
$$Z(A) = \sum_{X\...
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Pointless groups
This question now has two sequels, Pointless groups II (to which @R.vanDobbendeBruyn gave a counterexample for an infinite, imperfect field) and Pointless groups III, both using revised wording ...
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Is there a bijection between elements in algebraic closure of F2 and all bi-infinite periodic sequences made of 0 and 1, filling the properties below?
(And if so, how can I describe the "multiplication" on the sequence?)
We consider a bijection, denoted by $f$, from the algebraic closure of $\mathbb{F}_2$, named $\bar{\mathbb{F}_2}$, to ...
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Counting non-zero Gramians of Grassmanians over finite field
In case of $\mathbb{F}_{2}$, we can obtain the number of all reduced row echelon forms (so called Grassmannians) for some m$\times$n full rank matrices by the following gaussian polynomial,
$$
\binom{...
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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 ...
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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"...
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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^...
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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 ...
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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
$$\sum_{x \...
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A "semi-genetic" definition of addition and multiplication in the field $\operatorname{On}_p$?
Let $+,\cdot$ denote multiplication in $\mathbb{N}_0$. The addition and multiplication in $\operatorname{On}_p$ are denoted $\oplus, \otimes$.
Recursive definition of addition:
$$x \oplus y := ((x+y) \...
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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 (...
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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 ...