All Questions
Tagged with finite-fields nt.number-theory
220 questions
0
votes
1
answer
448
views
Geometric progression modulo p [closed]
Let $p\ge 11$ be a prime number, $n \ge 5$ be an odd positive divisor of $p-1$ and $s \in \mathbb Z_p$ such that $ord_p(s) = n$.
Is it true that the geometric progression $\{s^k\}_{k \in \mathbb Z_n}$...
- 169
1
vote
1
answer
802
views
Trace 0 and Norm 1 elements in finite fields
Let $\mathbb{F}_{q^\ell}/\mathbb{F}_{q}$ be the extension of finite filed $\mathbb{F}_{q}$, where $\ell$ be a odd prime and $(\neq q)$. Take $\zeta\in\mathbb{F}_{q^\ell}$. Does there exist different $...
- 255
14
votes
3
answers
2k
views
Collatz-like properties of finite fields
I was wondering what an equivalent of the Collatz conjecture might be for finite fields. In a Collatz sequence a number is moved down within a set $\{2^k n : k \in \mathbb{Z}^* \}$ for some odd $n$ or ...
- 261
5
votes
2
answers
571
views
Exceptional isomorphisms between finite simple Chevalley groups
Steinberg's "Lectures on Chevalley Groups"
https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf
contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...
- 5,050
15
votes
2
answers
1k
views
Can you use Chevalley‒Warning to prove existence of a solution?
Recall the Chevalley‒Warning theorem:
Theorem. Let $f_1, \ldots, f_r \in \mathbb F_q[x_1,\ldots,x_n]$ be polynomials of degrees $d_1, \ldots, d_r$. If
$$d_1 + \ldots + d_r < n,$$
then the ...
- 28.7k
6
votes
3
answers
912
views
Hypersurface missing just one point
Let $\mathbb F_q$ be a finite field and $n$ an integer.
What is the minimal degree $d = d(q,n)$ of a polynomial $f \in \mathbb F_q[X_1,\dots,X_n]$ such that the set $Z(f)$ of zeros of $f$ in the ...
- 26k
3
votes
0
answers
174
views
Carlitz factorials and Euler-like series
Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write $$n=n_0+n_1q+\cdots+...
- 3,998
2
votes
0
answers
124
views
Property of a derivative in global field
Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (https://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn'...
- 111
2
votes
0
answers
107
views
the least point on a variety over a finite field
Let $p$ be a large prime parameter and $V\subseteq \mathbb{P}^n_{\mathbb{F}_p}$ a variety defined over the finite field $\mathbb{F}_p$ with bounded degree and dimension (w.r.t. $p$). Assume that $V$ ...
- 3,362
2
votes
0
answers
119
views
A reference about a problem of the number of the rational points on a projective scheme
Let $X\hookrightarrow\mathbb{P}^n_{\mathbb{F}_q}$ be a pure dimensional projective scheme of dimension $d$. So we know a trivial estimate of the number of $X(\mathbb F_q)$ is that $\#X(\mathbb F_q)\...
- 403
5
votes
0
answers
108
views
Gaussian Hypergeometric Functions and Legendre Character
I was hoping somebody might be able to point me to a good reference on Gaussian hypergeometric functions defined over a finite field. the reason I'm interested is that I've encountered sums of the ...
- 51
5
votes
0
answers
205
views
Polynomials representing locally constant functions
Let $K$ be a finite field with $p$ elements.
(a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say ...
- 20.2k
7
votes
1
answer
927
views
Algebraic dynamics in finite fields
What is known about combinatorial structure of the rational maps of degree 2 over finite fields? From some general reasons I think it was studied. For being more specific, consider the field $\mathbb{...
- 109k
8
votes
0
answers
140
views
Order of zeros for sparse polynomials mod $p$
It is a fairly well known fact that sparse polynomials $f(x)$ cannot have large order zeros other than at $x=0$. If $f(x)=a_1x^{r_1}+\cdots+a_kx^{r_k}$ then at $c
\neq 0$, $f$ has a zero of order at ...
- 671
3
votes
3
answers
315
views
Minimal expression of 0 as a sum of kth powers in a finite field
Let $l=\min\{s\in \mathbb{N}|0\in s\cdot (\mathbb{F}_{p^n}^\times)^k\}$. Is any information known about this number already as a function of $k$? Any reference would be greatly appreciated!
- 203
7
votes
2
answers
330
views
When is Chevalley Warning's bound best possible?
Chevalley Warning's theorem (a form of) states that any homogeneous form over a finite field of degree $d$ in more than $d$ variables has a nontrivial zero in the field. However, for diagonal forms, ...
- 203
15
votes
0
answers
427
views
Counting abelian varieties over finite fields in a given isogeny class
Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the ...
- 148k
2
votes
1
answer
165
views
Proving inequation with ceilings in Finite Field of characteristic $p$
Take $ui = pt_i +j_i$ where $p$ is a prime number and $u(p-r) \equiv 1 $ $(\mbox{mod p})$ for positive integers $1 \le i, r, j_i\le p-1$ and $t_i \ge 0$. How can I prove that:
\begin{equation}
...
- 165
6
votes
1
answer
502
views
Finite field "contour" sum
Let $\Bbb{F}_q$ be a finite field. Choose a non-square $\delta \in \Bbb{F}_q^*$
and form the quadratic extension $\Bbb{F}_q\big( \sqrt{\delta} \, \big)$. For
an element $z \in \Bbb{F}_q\big( \sqrt{\...
- 2,137
1
vote
1
answer
142
views
What is the best known lower bound for: $\max_{2\leq i\leq p-1}(ord_n(i)) $?
Given an in integer $n$, and let $p$ be its smallest prime divisor (you can assume that $p$ is very large ). Let $ord_n(i)$ denotes the order of $i$ as an element of $\Bbb Z_n^*$ the multiplicative ...
- 223
4
votes
1
answer
382
views
Counting couples of square-free polynomials over finite fields
I have a curve defined by the following equations over the finite field $\mathbb{F}_q$ with $q=p^r$ with $p \geq 3$:
$$C_{h_1,h_2}:\begin{cases} y_1^2=h_1(t) & \\
y_2^2=h_2(t) &...
- 41
10
votes
0
answers
340
views
Geometric vs combinatorial motives over Spec Z
Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" ...
- 5,637
2
votes
0
answers
194
views
Number of common solutions of polynomial system
Let $ \mathbb{F}_p$ be a finite field and $\{f_j\}_{j=1}^{j=r} \subseteq \mathbb{F}_p[X_1,...,X_n]$ be a set of polynomials.
Let consider the system of equations:
$f_j(x_1,...,x_n)=0$ for $j = 1,......
- 2,576
3
votes
1
answer
485
views
Chebyshev polynomials factoring uniformly modulo all primes
Consider the Chebyshev polynomial of the first kind $T_n(x)$ and its factorization in $\mathbb F_p$ for a given prime $p$. Most often, this factorization is not uniform (meaning that the irreducible ...
- 13.4k
6
votes
1
answer
331
views
Which criteria for "uniformly splitting" polynomials?
Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials ...
- 13.4k
3
votes
1
answer
608
views
Representation of GL(n, F_p) over F_p, for n small
The question is related to this post
Representation theory of the general linear group over a finite prime field
However, I am asking for more detailed references for n small, for example, for n=2, ...
- 31
7
votes
1
answer
1k
views
Roots of a polynomial in a finite field related to Fermat's Last Theorem
In my class, we proved the following condition: define the polynomial $P_l(x)$ as
$$P_l(x) = \sum_{r=1}^{l-1}{\frac{1}{r}x^{l-1-r}}$$
Then if for all $a \in \mathbb{Z}/l\mathbb{Z}-\{0,1\},$ $P_l(x)$...
- 495
1
vote
0
answers
101
views
Points on the intersection of an affine quadric and cubic over a finite field
Are there absolute constants $N$ and $B$ such that the following is true?
Let $p>B$ be a prime. Let $q(x_0,\dotsc,x_n)$ and $c(x_0,\dotsc,x_n)$ be homogeneous of degree $2$ and $3$ with ...
- 3,142
6
votes
1
answer
372
views
A parity counting problem for subsets over finite fields
Let ${\mathbb F}_p$ be the prime field of $p$ elements and $b$ be an element in ${\mathbb F}_p$.
For a subset $T\subseteq {\mathbb F}_p$, define
$$Bias(T)=|N_e( {\mathbb F}_p,b)-N_o( {\mathbb F}_p,b)|...
- 121
1
vote
1
answer
323
views
Maximal separable extension of $\mathbb F_q((t))$
Let $K=\mathbb F_q((t))$. I want to prove that $K^{sep}$ is composite of $K^{sep}(p)$ and $K^{sep}(not \ p)$, where $K^{sep}(p)$ is maximal Galois extension of $K$ of exponent $p$, $K^{sep}(not \ p)$ ...
- 11
18
votes
5
answers
8k
views
Is $x^p-x+1$ always irreducible in $\mathbb F_p[x]$?
It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F_p$. (It is of course obvious that ...
- 17.9k
9
votes
1
answer
633
views
Motives over finite field not generated by hyperelliptic curves
So the question is that, over a finite field, does there exist an abelian variety $A$ for which there does not exist a generically one-to-one morphism from a hyperelliptic curve $C$ to $A$.
p.s. A ...
- 1,856
47
votes
3
answers
3k
views
A curious identity related to finite fields
To three elements $a_1$, $a_2$, $a_3$ in the finite field $\mathbb F_q$
of $q$ elements we associate the number $N(a_1,a_2,a_3)$
of elements $a_0\in \mathbb F_q$ such that the polynomial
$x^4+a_3x^3+...
- 17.9k
4
votes
0
answers
215
views
What is the function like when its Mobius inversion is $\sum_{w|r, (w,t)=1}\mu(w)q^{r/w}$?
Everyone, I am now reading a paper named The Irreducible Factors of $(cx+d)x^{q^m}-(ax+b)$ over $GF(q)$, http://qjmath.oxfordjournals.org/content/14/1/61.extract. And I’m confused with one of its ...
- 41
14
votes
3
answers
1k
views
Probability of coprime polynomials
Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let
$f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...
- 54.2k
3
votes
1
answer
394
views
When does a Bohr set have the right size?
Fix a set $
\Gamma\subset \mathbb F_p$, the field with $p$ elements and a parameter $\epsilon>0$. The Bohr set $B(\Gamma,\epsilon)$ consists of those $x$ for which $x\cdot \Gamma\subseteq[-\epsilon ...
- 133
1
vote
2
answers
472
views
Equations of elliptic curves
First part of question I have asked on mathoverflow already: https://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve
1) Let $E(\mathbb{F}_{q^2})$ is elliptic ...
- 2,576
4
votes
0
answers
144
views
Closest sumset to a set
Suppose $A$ is a subset of the finite field with $p$ elements. What is the best approximation of $A$ by a sumset $B+C$ in the sense that $|A\Delta (B+C)|$ is minimal? Of course if $B=A-x$ and $C=\{x\}$...
- 43
5
votes
1
answer
600
views
Are there Carlitz analogues of quadratic residues and reciprocity?
Let $q$ be a prime power. I will use the notations of Keith Conrad's Carlitz extensions paper (but I'll work over $\mathbb{F}_q$ rather than $\mathbb{F}_p$).
The most general question I'm asking here ...
- 33.8k
5
votes
0
answers
258
views
Transferring addition and multiplication over finite fields to $\mathbb{Z}$
It seems to me that the most basic wisdom on why many number-theoretic conjectures are hard is because the interplay between addition and multiplication is subtle and delicate (much of the lay chatter ...
- 15.4k
5
votes
2
answers
5k
views
A formula for a generator of the multiplicative group of $\mathbb{F}_p$ ?
Let $p$ be a prime. It is a common statement that the multiplicative group $(\mathbb{F}_p)^*$ of the prime field has no canonical generator. It is however no so easy to say exactly what this means, in ...
- 4,492
6
votes
2
answers
308
views
Polynomials over $\mathbb F_2$ without zeros in $\mathbb F_2$ having an inverse series with support of large density.
Does there exist a sequence $A_n=A_n(x)\in\mathbb F_2[x]$ over the field $\mathbb F_2$ of
two elements (represented by $0$ and $1$) such that $A_n(0)=A_n(1)=1$ and the inverse series $1/A_n=\sum_{j=0}...
- 17.9k
7
votes
0
answers
128
views
Sum of densities of support of $A$ and $A^{-1}$ for $A=1+\dots\in \mathbb F_2[[x]]$
Let $A=1+\dots\in\mathbb F[[x]]$ be a (multiplicatively) invertible series over the
field $\mathbb F_2$ of two elements. Writing $A=\sum_{n\geq 0}\alpha_n x^n$ and
$\frac{1}{A}=\sum_{n\geq 0} \...
- 17.9k
2
votes
1
answer
479
views
Small geometric progression modulo N
An problem related to integer factorization using the General Number Field Sieve is the following:
Let $N$ be a composite. Must there exist a 5-term geometric progression $\lbrace a_0,a_1,a_2,a_3,...
- 360
14
votes
3
answers
3k
views
Quadratic reciprocity and Weil reciprocity theorem
I was told that Weil reciprocity theorem (one has two meromorphic function $f,g$ on a complex curve $C$, so $\prod\limits_{x\in C} g(x)^{ord_xf}=\prod\limits_{x\in C}f(x)^{ord_xg} \ $ where $ord_xf$ ...
- 5,063
6
votes
3
answers
555
views
Source for embedding multiplicative group of an algebraic closure of a finite field?
It's easy to embed the (cyclic) multiplicative group of a finite field into the multiplicative group of $\mathbb{C}$ (or other algebraically closed field of characteristic 0): assign to a generator of ...
- 52.9k
20
votes
2
answers
2k
views
Sums of powers mod p
For prime $p > 7$ with $p-1=rs$, $r>1$, $s>1$, let $A=\{x^r|x \in \mathbb{Z}_p\}$ and $B = \{x^s|x \in \mathbb{Z}_p\}$. If $g$ is a primitive root mod $p$ then $A = \{0\} \cup \{g^{ir}|0 \leq ...
- 619
2
votes
0
answers
358
views
Counting points on an algebraic set over a finite field
Let $q=p^n$, for $p$ a prime. Let $C$ be an Artin–Schreier curve defined by $y^p-y=f(x)$ where $f(x) \in \mathbb{F}_{q}[x]$.
Let $C_g$ be $y^p-y=f(x)$ where $x \in g(\mathbb{F}_{p^{n}})$ for some $...
- 21
4
votes
4
answers
1k
views
Is there an analogue of finite fields for products of two prime powers?
The collection of prime powers can be characterized in the following way:
There is a field with $q$ elements if and only if $q$ is a prime power. Furthermore if it exists then this field is unique ...
- 1,723
14
votes
3
answers
3k
views
Truncated Exponential Series Modulo $p$: Deeper meaning for a Putnam Question.
Apparently B6 of the Putnam this year asked:
Suppose $p$ is an odd prime. Prove that for $n\in \{0,1,2...p-1\}$, at least $\frac{p+1}{2}$ of the numbers $\sum^{p-1}_{k=0} k! n^{k}$ are not divisble ...
- 11.4k