**2**

votes

**1**answer

224 views

### $\text{mod} \, p^2$ trace identity

Let $p$ be a prime, and let $\text{GL}_n \big( \Bbb{Z} / p^2 \Bbb{Z} \big)$ be the group of $n \times n$ invertible matrices over the ring $\Bbb{Z} / p^2 \Bbb{Z}$. Does there exist a positive integer $...

**5**

votes

**1**answer

440 views

### Full-rank rectangular matrices over GF(2)

Given positive integers $k$, $m$, $n$, let $A$ be an $m \times n$ matrix over $GF(2)$ constructed as follows. Let $X_1, \ldots, X_m$ be independent random subsets of $\{1,\ldots,n\}$ with cardinality ...

**36**

votes

**14**answers

5k 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 ...

**2**

votes

**2**answers

296 views

### On MDS code property

Is there a code that is Maximum Distance Separable and not isomorphic to Reed Solomon Codes? When is a MDS code isomorphic to Reed Solomon Code?
Is there an easy test? If so, could someone provide ...

**12**

votes

**3**answers

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

**3**

votes

**1**answer

229 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 ...

**4**

votes

**2**answers

709 views

### Quotient of $Z[x_1,…,x_n]$ by a maximal ideal is a finite field [duplicate]

I am seeing the proof of the Ax-Groethendieck theorem from commutative algebra and I have a problem. How can I prove that if $x_1,...,x_n$ are complex numbers and $I$ is a maximal ideal of $\mathbb{Z}[...

**1**

vote

**0**answers

166 views

### Finite fields: alternating sums of values of polynomials

Notation
In what follows let $p$ be a (odd, if needed) prime, $e$ a positive integer, $q = p^e$; $\mathbb{F}_q$ will denote a finite field with $q$ elements whose prime subfield will be denoted as $\...

**1**

vote

**3**answers

276 views

### Complete sets of functions

A (finite) set $S$ of boolean functions is called functionally complete if every boolean function can be presented as a finite composition of functions from $S$. For example, $\{ \neg,\wedge \}$ is ...

**10**

votes

**2**answers

597 views

### Seeking conceptual explanation of these nice bijections on roots of unity

I proved the following facts by unenlightening calculations. Since the statements are quite clean, I think there should be a conceptual explanation for them, which my proof certainly is not.
Let $q$ ...

**6**

votes

**1**answer

248 views

### Polynomiality of functions over residue rings

Suppose $\mathbb{Z}/m \mathbb{Z}$ is a residue ring for some $m \in \mathbb{N}$. If $m=p$ is a prime number then every function $f:\mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p \mathbb{Z}$ is a ...

**0**

votes

**1**answer

309 views

### Number of Minimal left ideals in the full matrix ring over a finite commutative local ring

Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?

**8**

votes

**1**answer

311 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 ...

**1**

vote

**2**answers

417 views

### Equations of elliptic curves

First part of question I have asked on mathoverflow already: http://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve
1) Let $E(\mathbb{F}_{q^2})$ is elliptic ...

**16**

votes

**1**answer

262 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 $g-I$ ...

**4**

votes

**2**answers

1k views

### Calculus over finite fields

$P(x,y,z)$ is a polynomial function on an algebraic surface $S$ in $F_{q}^{3}$. Suppose that the derivative of $P$ along any tangent vector of $S$ is zero. Can we say that $P$ is constant on $S$?
...

**3**

votes

**0**answers

106 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 $C=\{x\}$...

**5**

votes

**1**answer

473 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 ...

**3**

votes

**2**answers

406 views

### probability of having linearly independent sparse vectors over finite fields

Suppose that there are $i< N$ linearly independent $N$-dimensional vectors $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_i$ over a finite field $\mathbb{F}_q$, whose elements are denoted as $\{0,1,2,...

**5**

votes

**0**answers

225 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 ...

**0**

votes

**1**answer

345 views

### Existence of nontrivial roots of a homogeneous polynomial over a finite field in extension fields

Consider a homogeneous polynomial, f, of total degree n in n variables, with coefficients in a prime order finite field, GF(p).
Are there any general rules regarding the existence of nontrivial roots ...

**3**

votes

**1**answer

559 views

### Does there exist a polar decomposition of matrices over finite fields?

There exists a polar decomposition for matrices over the reals.
What I would like to know is if an analog has been shown for groups of matrices over finite fields. If not, it would be great to get ...

**7**

votes

**2**answers

513 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 ...

**10**

votes

**0**answers

270 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, $Pic(S)\simeq \mathbb Z$), ...

**9**

votes

**4**answers

1k 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 ...

**4**

votes

**1**answer

587 views

### Trivializing principal bundles on a curve over a finite field

This is related to my question Adelic description of moduli of $G$-bundles on a curve.
Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and $G$ a ...

**5**

votes

**1**answer

398 views

### More expanders?

Having received several exhausting answers to my recent question about
the expansion properties of a certain graph, I now wonder whether anything is
known on the following graphs of a similar nature:
...

**8**

votes

**4**answers

1k 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
$\...

**12**

votes

**2**answers

581 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 ...

**2**

votes

**0**answers

114 views

### Family with a fixed special fiber over finite fields

Let $X$ be a smooth projective variety over a finite field $\mathbb{F}_p$. What are the conditions for the existence of a projective variety $X'$ over $\mathbb{Q}_p$ such that $X$ is a special fiber ...

**4**

votes

**2**answers

2k 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 ...

**5**

votes

**2**answers

278 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}...

**3**

votes

**1**answer

1k views

### Order of an element in a finite field

Let $\mathbb F_p$ be the finite field of a prime order $p$, $f(x)\in \mathbb F_p[x]$ an irreducible polynomial, $E = \mathbb F_p[x]/\langle f(x)\rangle$ a finite extension of $\mathbb F_p$, $\lambda\...

**6**

votes

**0**answers

118 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} \...

**2**

votes

**2**answers

162 views

### linear independence of orbits via a set of transformations in char p

Let $T_1, \ldots, T_n \in GL(n,\mathbb{F}_p)$. Suppose for all $\vec{v} \in \mathbb{F}_p^n$ we have $\det (T_1 \vec{v}, T_2 \vec{v}, \ldots, T_n \vec{v}) = 0$. Now, let $k$ be a finite extension of $\...

**1**

vote

**1**answer

278 views

### Distribution of the powers of a primitive element of a finite field

What are known results regarding the distribution of the powers of a primitive element (generator of the multiplicative group) of a finite field? Specifically, compare the ordered list of ascending ...

**1**

vote

**2**answers

309 views

### When is PSU(2,q^2) = PSL(2,q) ?

The context for this question is from page 284 - 287 of Berger's paper: http://pdn.sciencedirect.com/science?_ob=MiamiImageURL&_cid=272332&_user=209810&_pii=S0021869398976785&_check=y&...

**4**

votes

**2**answers

483 views

### The Lang isogeny

Let $G$ be a connected commutative algebraic group over $\mathbb{F}_q$. If $\text{Fr}_q : G \to G$ denotes the $q$-Frobenius morphism, we define the Lang isogeny $L_q$ to be the endomorphism of $G$ ...

**0**

votes

**1**answer

106 views

### Probability of summing products of irreducible polynomials in a finite field to zero

Let $f(x), g(x), h(x)$ be randomly chosen irreducible polynomials over the finite field $GF(2^n)$.
What would the probability be for $\sum_{(i,j,k:i,j,k\in\mathbb{N},i+j+k=C)} f^i(x)g^j(x)h^k(x)=0$, ...

**4**

votes

**0**answers

287 views

### Tangent space of the moduli stack of Drinfeld modules

I am going through the proof of Thm 1.5.1 of Laumon, Cohomology of Drinfeld modular varieties, which says that a certain map of stacks is smooth. To prove this, Laumon considers the tangent space of a ...

**4**

votes

**0**answers

311 views

### Does the Riemann hypothesis for liftable varieties over a finite field imply the Riemann hypothesis for all varieties over a finite field

The Riemann hypothesis for varieties over a finite field has been proven by Deligne. Still I would like to ask the following question.
A variety $X$ over a finite field $k$ is liftable if there ...

**3**

votes

**0**answers

154 views

### Number of Inverse Pairs Modulo Prime $p$

Is there a result which gives a lower bound on the number of inverse pairs $(a, a^{-1})$ modulo prime $p$ lying in the interval $[1,t]$, where $t < p$?

**4**

votes

**1**answer

231 views

### Covering all, but $k$ points with affine subspaces

For non-negative integer $d\le n$ and $k\le 2^n$, how many affine subspaces of co-dimension $d$ are needed to cover all, but exactly $k$ elements of the vector space ${\mathbb F}_2^n$, and what are ...

**2**

votes

**1**answer

357 views

### Finding a subspace disjoint from a union of subspaces

Let $k$ be a finite field (I care about $\mathbb F_p$, especially $\mathbb F_2$) and let $V_1,...,V_N\subset k^n$ be subspaces.
I want to find a subspace $S\subset k^n$ such that $S\cap V_i=0$ for ...

**3**

votes

**1**answer

157 views

### Does the Hilbert polynomial determine the weight of the Galois representation associated to a variety

Let $X$ be a curve or an abelian variety (over a finite field). Then the Galois representation associated to $X$ via the etale cohomology of $X$ (in degree $1$) is integral of weight $1$ and its ...

**0**

votes

**0**answers

247 views

### Vanishing of motivic cohomology with finite coefficients in negative degrees

I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not.
STATEMENT:
Let $X$ be a smooth and projective scheme over a finite field $\...

**1**

vote

**0**answers

144 views

### Special values of zeta functions and extensions of base fields.

Let $X$ be a scheme of finite type over a finite field $k=\mathbb{F}_{q}$ of $q$ elements.
Then, one can define the zeta function $Z_{X/k}(T)$ of $X$ ovet $k$ as $\prod_{x\in |X|}\frac{1}{1-T^{deg_{k}...

**1**

vote

**1**answer

2k views

### Primitive $k$th root of unity in a finite field $\mathbb{F}_p$

I am given a prime $p$ and another number $k$ ($k$ is likely a power of $2$). I want an efficient algorithm to find the $k$th root of unity in the field $\mathbb{F}_p$. Can someone tell me how to do ...

**3**

votes

**1**answer

346 views

### Abelianized fundamental group of a curve over a finite field

Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and fix a geometric point $\overline{x} : \text{Spec } \overline{\mathbb{F}_q} \to X$. Then there ...

**4**

votes

**3**answers

2k views

### Number of solutions of an equation over finite fields

Does anyone know of any result that deals with the following problem of counting the number of solutions of a certain algebraic equation over a finite field?
Let $p$ be an odd prime and $(a,b,c,d)\in\...