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.
773
questions
4
votes
1
answer
206
views
Points on affine hypersurface over finite field
I am interested in the hypersurface $X\subset\mathbb{A}^4_{\mathbb{F}_{5^n}}$ defined by
$$
X = \{x^3 + 3xy^2 + z^3 + 3zw^2 + 1 = 0\}
$$
over a finite field $\mathbb{F}_{5^n}$ with $5^n$ elements. Via ...
- 5,888
2
votes
1
answer
188
views
Chinese remainder theorem for composition
Let $f(x) \in F_p[x]$ and I know $f(x)$ modulo two polynomials $\phi_1(x)$ and $\phi_2(x)$. What sort of information about $f$ modulo the composition $\phi_1(\phi_2(x))$ can I recover?
I'm looking ...
- 137k
6
votes
2
answers
399
views
Good and bad reduction for twists of algebraic curves
Suppose we have two curves $C/\mathbb{Q}$ and $C'/\mathbb{Q}$ which are twists of each other i.e. they are isomorphic over a field extension $K/\mathbb{Q}$.
Suppose that $C$ has good reduction at a ...
- 595
2
votes
1
answer
149
views
Automorphism of positive characteristic field
Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$.
I am interested in ...
- 49.6k
7
votes
2
answers
305
views
Representations of $\operatorname{Sp}(2g,\mathbb{Z}_3)$
Let $V$ be a $2g$-dimensional vector space over $\mathbb{Z}_3 := \mathbb{Z}/3\mathbb{Z}$. First, $\operatorname{Sp}(2g,\mathbb{Z}_3)$ acts on $\Lambda^2(V)$, and this decomposition is reducible, as ...
- 11.4k
2
votes
0
answers
186
views
Factorization of the polynomial $x^k + x^{k-1} + x^{k-2} + \cdots + x + 1$ in $\mathbb{F}_2[x]$ [closed]
Is anything known about the factorization of the polynomial $x^k + x^{k-1} + x^{k-2} + \cdots + x + 1$ in $\mathbb{F}_2[x]$?
When can it be factored, what are the irreducible factors, what are the ...
- 60.2k
5
votes
1
answer
304
views
Galois action on algebraic K-theory of finite fields
This might be well-known to experts. I was just teaching a course where we went through some parts of Quillen's theorem computing the higher algebraic K-theory of finite fields. Denote by $\mathbb F_q$...
- 8,719
1
vote
1
answer
282
views
Szemerédi–Trotter type theorem in finite field
This question is about the content of this paper by J. Bourgain, N. Katz, T. Tao.
In the final step (page 18) of the proof of Szemerédi-Trotter type theorem, we have already known
$$|A''+A''|\lesssim ...
- 11.8k
0
votes
1
answer
79
views
Algorithm to find a number B with same modulus as A with prime P and specific binary positions set to zero
Given a prime $P$, an integer $A$ $(0\leq A<P$), and a set of legal positions (encoded as a binary mask $\text{mask}$), is there an efficient algorithm to find a number $B$ that has the same ...
- 5,169
2
votes
1
answer
146
views
On the estimate for the mixed 3-dimensional hyper-Kloosterman sum
There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum:
For any positive integer $n$ not divisible by $p$, how to prove
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p}
\...
- 137k
3
votes
0
answers
112
views
Effective Lang-Weil bounds for higher codimension varities
The Lang-Weil bounds imply that for a geometrically irreducible variety $X$ of dimension d over $\overline{\mathbb{F}_p}$ we have,
$$|N_q(X)-q^{d}|\le O(q^{d-1/2}),$$
where $N_q(X)$ is the number of $\...
- 145
7
votes
1
answer
1k
views
Given $v,w$ primes of $k$, is there $K/k$ so $K_v\cap\Bbb Q^\text{cycl}=K_w\cap\Bbb Q^\text{cycl}=K\cap\Bbb Q^\text{cycl}$?
For any field $k$, let $\mu(k)$ denote the roots of unity in $k$. Now let $k$ be a number field and let $v, w$ be non-archimedean primes of $k$ with distinct residual characteristics. Does there ...
- 11.9k
4
votes
1
answer
330
views
Polynomial that is not always a square over $\mathbb{Z}_p$
Let $p > 3$ be prime. Is is true that there exists $x \in \mathbb{Z}_p$ such that
$$
(1+x^2)^3-1
$$
is not a square in $\mathbb{Z}_p$? In particular, when $-1$ is not a square in $\mathbb{Z}_p$, ...
- 77.5k
0
votes
0
answers
107
views
Cubic monic polynomial over z_p
Let
$$
f_{a}(x)=x^3+(u-2-a)x^2+ax+1,
$$
where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all $a\in\mathbb{Z}_p$ such that $f_{a}(x)$ factor linearly. Then what is the cardinality ...
CommunityBot
- 1
13
votes
2
answers
721
views
Isomorphism of varieties in $\mathbb{C}$ implies isomorphism over finite fields
Suppose that $X$ and $Y$ are algebraic varieties over $\mathbb{Z}$ (add your favourite hypotheses, like smooth or affine if needed). Denote by $X_k$ and $Y_k$ their base-change to varieties over a ...
- 4,279
1
vote
0
answers
65
views
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?
- 11
4
votes
0
answers
210
views
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 (...
- 4,353
5
votes
1
answer
563
views
Aschbacher classes and $\mathbb{F}_p$-subspace stabilizers in classical linear groups
I am reading the Kleidman–Liebeck book ("The subgroup structure of the finite classical groups") which is about the Aschbacher classification of maximal subgroups of the classical almost ...
- 11.4k
1
vote
1
answer
430
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 $p$...
CommunityBot
- 1
1
vote
0
answers
222
views
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 ...
- 367
6
votes
1
answer
307
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 ...
- 11.4k
7
votes
0
answers
175
views
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)|\...
- 25.6k
4
votes
2
answers
414
views
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 ...
- 121
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 ...
- 137k
0
votes
0
answers
171
views
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 ...
- 60.2k
2
votes
0
answers
154
views
Determining the multiple solutions for $\mathrm{GF}(2)$ discrete logarithms of polynomials with partially known coefficients
I have an LFSR, essentially $x^k \bmod p(x)$ for some characteristic primitive polynomial of degree $N$ with coefficients in $\mathrm{GF}(2)$, as outlined in Clark and Weng's article: it has a period $...
- 60.2k
1
vote
0
answers
140
views
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 ...
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} = ...
- 1,801
2
votes
1
answer
297
views
Papers on distribution of high order elements over $\mathbb{F}_p$
I am interested in knowing about the distribution of exponentially high order elements in $\mathbb{F}_p$. To be precise let $s$ be of the order $\frac{p}{\log^{k}(p)}$ for some fixed $k$ and integer. ...
- 60.2k
2
votes
1
answer
385
views
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^...
3
votes
0
answers
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 ...
- 11.8k
10
votes
8
answers
1k
views
Classifications of finite simple objects
I'm curious to know if other classifications are known of "finite simple structures" in the same spirit of the monumental classification of finite simple groups. Here I mean "...
- 4,618
2
votes
1
answer
172
views
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}...
- 17.4k
0
votes
0
answers
117
views
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 ...
- 13.7k
4
votes
1
answer
545
views
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 $C_n\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 ...
2
votes
0
answers
38
views
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 $...
- 11.4k
2
votes
0
answers
112
views
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 ...
- 3,024
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 ...
- 17.4k
1
vote
0
answers
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})^...
- 543
4
votes
1
answer
220
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 $...
- 151k
18
votes
2
answers
928
views
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$-...
- 42.8k
6
votes
0
answers
243
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 ...
- 61
0
votes
0
answers
89
views
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}...
- 1
5
votes
1
answer
282
views
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|,
$$
with $X^g$ being the set of ...
- 151k
7
votes
2
answers
397
views
Sum of two $n$th powers in finite fields
Let $q$ be a prime power, let $n$ be a positive integer and let $\mathbb{F}_q$ be the finite field of cardinality $q$. I have some computational evidence that the set $$\{x^n+(-1)^nay^n:x,y\in\mathbb{...
- 11.4k
8
votes
1
answer
331
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 ...
- 2,643
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 ...
- 88.8k
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 \...
- 221
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 ...
- 1,744
6
votes
0
answers
261
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 ...
- 7,646