The finite-fields tag has no wiki summary.

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### Cyclic order relation in Zn

The ring Zn:={0,1,..,n-1} under addition and multiplication modulo n.
Suppose a,b,c,x $\in$ Zn are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?

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**9**answers

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### 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.
...

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votes

**1**answer

291 views

### additive structure in a small multiplicative group of a finite field?

Let $p$ be a prime. Given a positive integer $n$, does there exist a
$\beta$ in an extension of $F_p$ such that
1) If $F_p[\beta] = F_{p^N}$, then $N > n^n$; ( $\beta$ lies in a high extension)
...

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**1**answer

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### Inverse for a permutation over GF(2)

Given a permutation $f: \{0,1\}^n \rightarrow \{0,1\}^n$ as $n$ polynomials over $GF(2)$ how to get formulas for the inverse permutation $f^{-1}$?
I am interested in the answer to the previous ...

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votes

**1**answer

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### 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 ...

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vote

**3**answers

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### How to factorize X^n - 1 in Z/pZ?

How do I factorize a polynomial $X^n - 1$ over $\mathbb{F}_p$? In particular I need to find factors of the polynomial $X^{3^3 - 1} - 1 = X^{26} - 1$ over $\mathbb{F}_3$.

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votes

**1**answer

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### 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 ...

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**3**answers

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### Complex powers in finite fields

Is it possible to compute complex powers in finite fields? Given a $\in \mathbb{F}_p$ ($p$ prime), how can one compute $a^i$ per example?

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votes

**3**answers

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### A quadratic form

Let $q$ be a power of 2. Let $P$ be the set of polynomials in
$F_q [x]$ of degree d or less.
Let $\mathbb{Z}$ be the ring of integers. For any $f \in P$, let $\psi(f)$ be the
number of distinct ...

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votes

**1**answer

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### 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?

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**2**answers

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### Computer program to solve a system of polynomial equations over a finite field

I have a set of polynomial equations for which I want to know the solutions (actually really the number of solutions). It would be great if I could get a computer to do it, but I'm not sure exactly ...

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**3**answers

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### 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 ...

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votes

**2**answers

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### Elementary theory of finite fields

I read on Ax's article that the elementary theory of finite fields is decidable if one assumes the continuum hypothesis to be true. What about if one assumes the hypothesis to be false?

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votes

**3**answers

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### 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 ...

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votes

**2**answers

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### Weil Conjectures for Grassmannians

To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?

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votes

**1**answer

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### Counting solutions to x^{p+1}=y^4 in a finite field

I need to compute the number of solutions to the equation $x^{p+1} = y^4$ in the field with $p^2$ elements (for p sufficiently large). The form of the equation suggests to me that the solution would ...

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votes

**2**answers

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### 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 ...

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votes

**2**answers

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### Induction from split and non-split tori for GL_2 over a finite field

Let k be a finite field, G the k-points of GL_2, T1, T2 the k-points of the split and non-split tori of G.
Then the G-representations C[G/T1] and C[G/T2] are almost the same.
More precisely, they ...

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votes

**4**answers

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### 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 ...

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### 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 ...

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### 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 ...

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### 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.

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### 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 ...