All Questions
16 questions
14
votes
1
answer
285
views
Lower bounds for class number of function fields with fixed $q$, growing $g$
Let $X$ be a smooth project curve of genus $g$ over the finite field with $q$ elements. Let $h$ be $\# \mathrm{Pic}^0(X)(\mathbb{F}_q)$. Weil showed that $h \geq (\sqrt{q}-1)^{2g}$. Lachaud and Martin-...
- 156k
9
votes
0
answers
462
views
Who realized the finite fields $\mathbb F_{p^n}$ first? Gauss or Galois?
Let $p$ be a prime, and let $n$ be a positive integer. The finite field $\mathbb F_{p^n}$ is often called a Galois field and denoted by $\mathrm{GF}(p^n)$ by researchers on coding theory.
On the other ...
- 15.6k
8
votes
2
answers
563
views
Distribution of primitive roots, as p varies
For a prime number $p$, let $\Phi(p)$ be the subset of $\{ 1, 2, \ldots, p-1 \}$ consisting of primitive roots modulo $p$. (Thus $\# \Phi(p) = \phi(p-1)$, where $\phi$ denotes the totient.)
I am ...
- 13.3k
8
votes
1
answer
355
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 ...
- 7,193
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
6
votes
1
answer
788
views
Exponential sums over finite fields with even characteristic
I am looking for an elementary evaluation (if one exists) of the exponential sum
$$
G_r(a,b) = \sum_{x \in \mathbb{F}_{2^r}} \psi(ax^2 + bx),
$$
where $a,b \in \mathbb{F}_{2^r}^*$ are both units, $\...
- 197
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
6
votes
0
answers
176
views
Fundamental lemma of sieve theory in function fields
Is there any literature concerning the fundamental lemma of sieve theory in $\mathbb{F}_q[T]$?
In integers there are various versions of the lemma (bases on different sieves); I would be happy with ...
- 14.6k
5
votes
1
answer
223
views
Intrinsic characterisation of a class of rings
This may be well known, but I was unable to find an answer browsing literature. Let us temporarily call a commutative (unital) ring $R$ an O-ring if there exists an integer $n \ge 1$, a local field of ...
- 6,224
4
votes
1
answer
291
views
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
$$\#\{\text{prime ...
- 43
3
votes
3
answers
315
views
Minimal expression of 0 as a sum of kth powers in a finite field
Let $l=\min\{s\in \mathbb{N}|0\in s\cdot (\mathbb{F}_{p^n}^\times)^k\}$. Is any information known about this number already as a function of $k$? Any reference would be greatly appreciated!
- 203
2
votes
1
answer
192
views
A Vandermonde-type system
For a prime $p$ and $a_1,\dotsc,a_n\in\mathbb F_p^\times$, consider the system of equations
$$ \begin{cases}
\begin{align}
a_1 + \dotsb + a_n &= 0 \\
a_1x_1 + \dotsb + a_nx_n &...
- 23k
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. ...
- 335
2
votes
0
answers
157
views
On hypergeometric functions over finite fields
Let $\mathbb{F}_q$ be a finite field of $q$ elements. Let $A,B,C,\cdots$ denote the multiplicative characters over $\mathbb{F}_q$, and let $\overline{A}$ denote the inverse of $A$, i.e., $A(x)\...
- 93
2
votes
0
answers
145
views
On some rational points on an elliptic curve over finite field
Let $p\equiv3\pmod4$ be a prime. We consider the elliptic curve $E$ over the finite field $\mathbb{F}_p$
(in affine coordinates) defined by
$$y^2=x^3+x.$$
Clearly the discriminant of $E$ is $-2^6$. ...
user125345
2
votes
0
answers
186
views
Dyadic models in number theory and "spillover"
In a classic blog post, Tao discusses the appearance of "dyadic models" in various guises in various areas of math. The number-theoretic version of the idea is to study polynomials over a ...
