The finite-fields tag has no wiki summary.

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### 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 ...

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votes

**2**answers

417 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 ...

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

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votes

**4**answers

964 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 ...

**3**

votes

**1**answer

520 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 ...

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votes

**1**answer

381 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:
...

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votes

**4**answers

952 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
...

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votes

**2**answers

435 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 ...

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votes

**0**answers

110 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 ...

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votes

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1k 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 ...

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votes

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273 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 ...

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votes

**1**answer

835 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 $\mathbb F_p$, ...

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votes

**0**answers

107 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} ...

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votes

**2**answers

158 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 ...

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vote

**1**answer

246 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 ...

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**2**answers

279 views

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

The context for this question is from page 284 - 287 of Berger's paper: ...

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**2**answers

392 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 $G$ ...

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votes

**1**answer

96 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 $\sum_{(i,j,k:i,j,k\in\mathbb{N},i+j+k=C)} f^i(x)g^j(x)h^k(x)=0$, ...

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254 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 ...

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**0**answers

304 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 ...

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**0**answers

147 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 $t < p$?

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**1**answer

301 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 ...

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votes

**1**answer

149 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 ...

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219 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 ...

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136 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 ...

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vote

**1**answer

1k 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 ...

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306 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 ...

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1k 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 ...

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**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 ...

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**0**answers

209 views

### Small geometric progression modulo N

An problem related to integer factorization using the General Number Field Sieve is the following:
Let $N$ be a composite. Must there exist a 5-term geometric progression $\lbrace ...

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### computer algebra system for polynomial algebras over finite fields

Is there a computer algebra system that can do arithmetic over polynomial algebras over finite fields where I can specify the extension?
Exempli gratia, if $f(x), g(x) \in ...

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votes

**1**answer

146 views

### trace cotrace Matrix

Hello I want to know whats mean (trace) cotrace matrix.
In the context, mapping a matrix (t x n) $\in$ GF($2^m$) to a cotrace matrix (tm x n) $\in$ GF($2$)?

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369 views

### Finite Field Varieties and the de Rham Complex of Kähler Differentials

In an answer to a previous question I asked, where the Kahler differentials of a variety over a finite field were discussed, it was stated that:
You can certainly define de Rham cohomology using ...

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922 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 ...

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**2**answers

392 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 ...

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200 views

### Given g1(x), g2(x) minimize over p(x) Hamming weight of [p(x)g1; p(x)g2(x) ] ? (Or how to find minimal distance of convolutional code?)

Fix polynoms g1(x), g2(x) over F_2[x].
Question: How to find minimum over polynoms p(x) of the:
HammingWeight(p(x) g1(x) ) + HammingWeight(p(x) g2(x) ) ?
By HammingWeight of polynom I mean number ...

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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 ...

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votes

**1**answer

252 views

### Special polynomials over finite fields

My field of research is coding theory and I am working on cyclic codes. During my research, I tackled an algebraic problem. After some simple definitions, I asked my question. I will appreciate any ...

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259 views

### Combinatorial Interpretation of an Extension of Gaussian Polynomials

It is well-known that the Gaussian polynomial (or Gaussian coefficient, q-binomial coefficient) $\binom{n}{k}_q$ counts the number of $k$-dimensional subspaces of an $n$-dimensional vector space over ...

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**1**answer

190 views

### How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

Let us consider polynoms over $F_2$.
Consider the linear SUBSPACE of polynoms divisible by $x^{16}+x^{12}+x^5 +1$ and of degree less or equal $N$ (e.g. 40).
Question: How many k-nomials belong to ...

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vote

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134 views

### Number of solutions to $mx^2+ny^2 \equiv k\pmod{p}$

I need a reference for the result which gives the number of solutions to the congruence $mx^2+ny^2 \equiv k\pmod{p}$. This result seems to be something that would be discussed in Gauss' ...

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922 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 $ord_xf$ ...

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164 views

### Matrices whose range is equal to the column set [closed]

Is there such a thing?
I suppose there are two cases in which they may exist: over a finite field or infinite matrices (but let's stick to countable matrices then).

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297 views

### Matching a binary matrix

Given a MxN 0-1 matrix D, with the property that
both M and N are odd numbers
its row sums and column sums in the $\mathbb{Z}_2$ field are all equal to the same number (0 or 1).
How do we find M ...

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355 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 ...

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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 ...

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**1**answer

446 views

### Finite Field Grassmannians as Homogeneous Spaces

For the real Grassmannian Gr$(N,k)$ we have the well-known isomorphism
$$
\text{Gr}(N,k) = O(N)/(O(k) \times O(N-k))
$$
For the complex case, we have
$$
\text{Gr}(N,k) = U(N)/(U(k) \times U(N-k))
$$ ...

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votes

**3**answers

583 views

### homogeneous polynomials over a finite field

Let $P(x_1, \ldots , x_n)$ be a homogeneous polynomials over a finite field with $q$ elements. Is there any way to count all the roots of $P$?

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871 views

### Vandermonde matrices and general position

I was wondering if it is known whether a Vandermonde matrix over a sufficiently large finite field is in general position with respect to intersections of subspaces spanned by subsets of
columns, i.e. ...

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votes

**1**answer

391 views

### Probability of a set of random vectors over finite field being a spanning set

Suppose I have a set of random vectors $f(a_1, \ldots, a_\ell) := (v_1, \ldots, v_m) \subset F_p^n$, $m \ge n$, given by a matrix valued polynomial function $f$, where the $a_i$'s are independent, ...