The finite-fields tag has no wiki summary.

**34**

votes

**2**answers

806 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
...

**23**

votes

**9**answers

2k views

### How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings,
fields, graphs, partial orders, etc.
...

**22**

votes

**1**answer

1k views

### How does Tate verify his own conjecture for the Fermat hypersurface?

This question is about Tate's 1963 paper "Algebraic Cycles and Poles of Zeta Functions". Here he announces a conjecture (now known as "the Tate conjecture") which states that certain classes in the ...

**21**

votes

**3**answers

595 views

### A hypersurface with many points

Ok, it's time for me to ask my first question on MO.
Consider the affine curve $Y+Y^q=X^{q+1}$ over the finite field $\mathbf{F}_q$. It's interesting because it has the largest number of points over ...

**20**

votes

**2**answers

833 views

### Does the etale fundamental group of the projective line minus a finite number of points over a finite field depend on the points?

Clearly the etale fundamental group of $\mathbb{P}^1_{\mathbb{C}} \setminus \{a_1,...,a_r\}$ doesn't depend on the $a_i$'s, because it is the profinite completion of the topological fundamental group. ...

**19**

votes

**2**answers

1k 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 ...

**18**

votes

**8**answers

2k views

### Fantastic properties of Z/2Z

Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...

**17**

votes

**3**answers

874 views

### Non-split extensions of $GL_n(F_q)$ by $F_q^n$ ?

A very naive question :
I just learned that there is a non-split extension of $GL_3(F_2)$ by $F_2^3$ (with standard action). It can be realized as the subgroup of the automorphism group $G_2$ of ...

**16**

votes

**4**answers

906 views

### A mixing property for finite fields of characteristic $2$

In connection with this MO post, here is a question somewhat implicitly contained in a joint paper of S. Kopparty, S. Saraf, M. Sudan, and myself.
Let ${\mathbb F}$ be a finite field, and suppose ...

**15**

votes

**1**answer

206 views

### Do mutually dual finite vector spaces have the same orbit cardinalities under a linear group action?

Let $G$ be a finite group acting linearly on a finite dimensional vector space $V$ over a finite field. By Burnside's lemma,
$$
|V/G| = \frac 1{|G|} \sum_{g\in G} q^{\dim(ker(g - I))}.
$$
Since $g-I$ ...

**13**

votes

**0**answers

448 views

### Noether-Deuring for injections and surjections?

Noether-Deuring theorem (not in the strongest form, but in the one I usually need):
Let $L\diagup K$ be a field extension. Let $A$ be a $K$-algebra which is finite-dimensional as a vector space over ...

**12**

votes

**4**answers

1k views

### Exhibit an explicit bijection between irreducible polynomials over finite fields and Lyndon words.

Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree ...

**12**

votes

**1**answer

485 views

### Does $\mathbb{K}[G]\simeq\mathbb{K}[H]$ for some field $\mathbb{K}$ of characteristic $p$, imply $\mathbb{F}_p[G]\simeq\mathbb{F}_p[H]$?

Due to the first (and very helpful) answer I received, I've reformulated the question a little: $G$ and $H$ are now assumed to be $p$-groups.
Let $\{p}$ be a prime, and let $\mathbb{F}_p$ be the ...

**12**

votes

**1**answer

708 views

### A family of hypersurfaces with many points

This question is a sequel to an earlier question, which asked about the zeta function of a certain affine variety over a finite field $k$. The unusual thing about this variety is that it had the ...

**11**

votes

**3**answers

2k 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 ...

**11**

votes

**2**answers

343 views

### An expander (?) graph

For a prime $p$, consider the graph on the vertex set ${\mathbb F}_p$, in
which every vertex $z$ is adjacent to $z\pm 1$ and also to $z^{-1}$ (unless
$z=0$). I was told that this graph is known to be ...

**11**

votes

**4**answers

521 views

### Behaviour of Zeta-function under Finite Morphism

Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of ...

**10**

votes

**5**answers

784 views

### Is $x^p-x+1$ always irreducible in $F_p[x]$?

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 ...

**10**

votes

**3**answers

403 views

### Local-globalism for similar matrices?

My background on number theory is very weak, so please bear with me...
Given two matrices $A$ and $B$ in $\mathbb{Z}^{n\times n}$. Assume that for every prime $p$, the images of $A$ and $B$ in ...

**10**

votes

**3**answers

627 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 $f_2 = x^{m_2} + (1+x)^{n_2}$ ...

**10**

votes

**3**answers

936 views

### Can we always find such an irreducible polynomial of degree n where degree(p(x)-x^n)<= n/2?

Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum_{i=0}^n a_i x^i$ where $a_i\in GF(2)$ and $a_n=1$.
Can we always find such an irreducible ...

**10**

votes

**3**answers

854 views

### Periodic orbits and polynomials

There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system.
fact 1 Consider the "tent map" ...

**9**

votes

**6**answers

1k views

### Proofs in the same vein as Ax-Grothendieck

I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...

**9**

votes

**2**answers

737 views

### Multiplicative order of zeros of the Artin-Schreier Polynomial

This question was asked on NMBRTHRY by Kurt Foster:
If $p$ is a prime number and $\mathbb{F}_p$ the field of $p$ elements, the zeroes of the Artin-Schreier polynomial
$x^p - x - 1 \in ...

**9**

votes

**2**answers

810 views

### A comprehensive overview of finite fields

I've read numerous introductions to finite fields, but I feel like my intuition about them is fairly lacking. Considering that finite fields are the the most "inert" objects in algebraic geometry, I ...

**9**

votes

**1**answer

817 views

### All polynomials over a finite field are sums of $2$ square-free polynomials

I quote Proposition 2.3, page 14 lines -3 and -4 of Michael Rosen's book
Number Theory in Function Fields:
Let $b_n$ be the number of square-free monics in $A= \mathbb{F}_q[t]$ of degree $n.$
Then ...

**9**

votes

**1**answer

254 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 ...

**9**

votes

**0**answers

224 views

### Noether-Lefschetz over finite fields

The classical Noether-Lefschetz theorem asserts the following: Over the complex numbers, a very general surface $S\subset \mathbb{P}^3$ has Picard number 1 (that is, $Pic(S)\simeq \mathbb Z$), ...

**8**

votes

**4**answers

901 views

### The “interplay” between additive and multiplicative structure in a field

A field is an ordered triple $(F, +,\cdot)$ of a set $F$ and binary operations $+,\times$ on $F$ such that $(F,+)$ and $(F\backslash 0,\times)$ are abelian groups satisfying the distributive laws
...

**8**

votes

**4**answers

1k views

### Equivalent Statements of Riemann Hypothesis in the Weil Conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with q elements says that: the eigenvalues of ...

**8**

votes

**2**answers

370 views

### A mixing property of linear map over finite fields

Let $F$ be a finite field of odd size $q$, and $\phi_0 : F \mapsto F$ be any map from $F$ to itself. For each $a \in F$, set $\phi_a : x \in F \mapsto \phi_0 (x) + ax $.
When $\phi_0 : x \mapsto x^2 ...

**8**

votes

**1**answer

546 views

### Restriction theorems over finite fields

A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the ...

**8**

votes

**1**answer

497 views

### Number of irreducible polynomials with some coefficients fixed over a finite field

I am interested in the following problem: I have a finite field $F_q$, two positive integers
$n>m$ and elements $a_1,...,a_m\in F_q$. How many of the polynomials
...

**8**

votes

**1**answer

284 views

### A Balog-Szemeredi-Gowers-type question

A nice Lemma due to Konyagin asserts that for any subset $B \subset \mathbb{F}_p$ holds
$$
|B.B - B.B + B.B - B.B + B.B - B.B| \geq \frac{1}{2}\min\{p, |B|^2 \},
$$
where the standard notation for the ...

**8**

votes

**2**answers

447 views

### Invariant functor for admissible representations of reductive groups over local fields

Hello,
I have a question concerning a certain functor between represention categories. I'm rather sure this is already known, but I could not find a reference.
Let $F$ be a local non-archimedean ...

**8**

votes

**0**answers

883 views

### roots of quadratic forms

This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not ...

**7**

votes

**3**answers

808 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 $ord_xf$ ...

**7**

votes

**2**answers

900 views

### What is an example of a smooth variety over a finite field F_p which does not lift to Z_p?

Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.

**7**

votes

**4**answers

1k views

### Orthogonal Groups over finite fields

Hello
Let $\mathbb{F}_p$ be the finite field with $p$ elements. One can show that over finite fields, there are just two non-degenerate quadratic forms.
So here I want to pick
any non-degenerate ...

**7**

votes

**2**answers

463 views

### Richardson varieties over finite fields

Let me start with some background to set the notation before I ask my question.
Let G be a semisimple algebraic group over some algebraically closed field K, and suppose we have fixed a Borel ...

**7**

votes

**1**answer

311 views

### algorithm for calculating the Chow groups of a variety over a finite field

Is there an algorithm for calculating the Chow groups of a variety over a finite field?
It is know that $H^{2i,i}_\mathrm{mot}(X,\mathbf{Z}) = CH^i(X)$. In how many cases does this help us?

**7**

votes

**2**answers

492 views

### Gelfand-Tsetlin bases for Lie groups over finite fields

There is a method of constructing representations of classical Lie algebras via Gelfand-Tsetlin bases. It has also been applied to Symmetric groups by Vershik and Okounkov. Does anybody know of any ...

**7**

votes

**1**answer

694 views

### étale cohomology with G_m coefficients

Most calculations of étale cohomology in Milne's book deal with constructible or torsion sheaves. Are there references where the cohomology of varieties with $\mathbf{G}_m$ coefficients are ...

**7**

votes

**0**answers

171 views

### A strong sum-product “for translates” in finite fields

In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...

**6**

votes

**1**answer

582 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, ...

**6**

votes

**2**answers

378 views

### How do the number of plane curves over a finite field of a fixed genus increase with the degree?

Let $k$ be a finite field. We define $N(d,g)$ to be the number of plane curves $f(x,y)$ defined over $k$ of degree $d$ with (geometric) genus $g$. If $D(d) := (d-1)(d-2)/2$ (the maximum possible ...

**6**

votes

**2**answers

331 views

### Dimension of incomplete matrix over finite fields.

Hi,
Suppose one has an incompletely specified $2^n \times 2^n$ matrix over some fixed finite field $\mathbb{F}_{p^k}$. In fact, one knows that the diagonal entries are zero and all other entries are ...

**6**

votes

**3**answers

321 views

### On the rank of a matrix $S$ with coefficients in $\mathbb F_{2^m}$

Let $\mathbb F$ be a finite field with characteristic 2 and let $S \in M(2k, 2k, \mathbb F)$ be the matrix defined as follows
$$ S=\left[\begin{array}{ccccccc}
0 & ...

**6**

votes

**2**answers

377 views

### request sources about self-dual cyclic codes

X. Kai and S. Zhu in the paper "On cyclic self-dual codes", AAECC, vol. 19, pp. 509-525, 2008, at page 510 in line 6 said that, It is well Known that there are no cyclic self-dual codes over $F_q$ ...

**6**

votes

**2**answers

1k views

### Algorithms to find irreducible polynomials of a given degree

I need to know what are the efficient algorithms to find all the irreducible polynomials of a given degree, say $d$ over a given finite field, say $\mathbb{F}_{p^n}.$
One way is to factorize the ...