# Tagged Questions

The finite-fields tag has no wiki summary.

**35**

votes

**2**answers

849 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
...

**28**

votes

**14**answers

3k 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 ...

**24**

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

**2**answers

917 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. ...

**21**

votes

**3**answers

602 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 ...

**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 ...

**17**

votes

**4**answers

914 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 ...

**17**

votes

**3**answers

890 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

**1**answer

213 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 $g-I$ ...

**16**

votes

**3**answers

1k views

### Classification of rings satisfying $a^4=a$

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...

**13**

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 ...

**13**

votes

**0**answers

457 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

**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 ...

**12**

votes

**1**answer

493 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

712 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

**2**answers

371 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

**3**answers

878 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" ...

**11**

votes

**4**answers

528 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

807 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

405 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

662 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

947 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

**2**answers

268 views

### Is there a proof of Warning's Second Theorem using p-adic cohomology?

Let $\mathbb{F}_q$ be a finite field, $n \in \mathbb{Z}^+$, and $f_1,\ldots,f_r \in \mathbb{F}_q[t_1,\ldots,t_n]$ with $\operatorname{deg}(f_i) = d_i$. Put $d = \sum_{i=1}^n d_i$ and suppose $d< ...

**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

744 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

815 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

842 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

293 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

234 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

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

384 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

559 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

504 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

291 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

460 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

156 views

### Algebraic dependency over $\mathbb{F}_{2}$

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$
such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall ...

**8**

votes

**0**answers

917 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

**4**answers

916 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
...

**7**

votes

**3**answers

846 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

927 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

475 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

317 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

493 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

707 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

**4**answers

812 views

### Books on advanced galois theory

I have been studying galois theory on my own and find it very fascinating. I have gone through Ian Stewarts book: http://www.amazon.co.uk/Galois-Theory-Third-Chapman-Mathematics/dp/1584883936. I am ...

**7**

votes

**0**answers

186 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 ...

**7**

votes

**0**answers

333 views

### On the number of polynomials that divide $x^{q-1}-1$ in some subspaces of $\mathbb F_q[x]$

Let $\mathbb F_q$ be a finite field (where $q$ is in general a power of a prime), and let $e, k$ be positive integers with $k \leq e < q-1$. Let $f_0(x), \ldots,
f_k(x) \in \mathbb F_q[x]$ be ...

**6**

votes

**7**answers

550 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 "classification" in the ...