The finite-fields tag has no usage guidance.

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273 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 ...

**1**

vote

**1**answer

191 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

**2**answers

139 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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votes

**3**answers

974 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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votes

**2**answers

165 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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**0**answers

301 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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votes

**3**answers

360 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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**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 ...

**4**

votes

**1**answer

479 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

598 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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**2**answers

888 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

394 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, ...

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votes

**0**answers

293 views

### Counting points over over an algebraic set over finite field.

Let $q=p^n$. Let $C$ be an Artin schierer curve defined by $y^p-y=f(x)$ where $f(x) \in \mathbb{F}_{q}[x]$.
Let $C_g$ be $y^p-y=f(x)$ where $x \in g(\mathbb{F}_{p^{n}})$ for some $g \in ...

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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 ...

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**2**answers

397 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 $F_q$ ...

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votes

**1**answer

299 views

### Aschbacher classes and $\mathbb{F}_p$-subspace stabilizers in classical linear groups

I am reading the Kleidman-Liebeck book ("The subgroup structure of the finite classical groups") which is about the Aschbacher classification of maximal subgroups of the classical almost simple ...

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votes

**4**answers

916 views

### Is there an analogue of finite fields for products of two prime powers?

The collection of prime powers can be characterized in the following way:
There is a field with $q$ elements if and only if $q$ is a prime power. Furthermore if it exists then this field is unique ...

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**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 ...

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**3**answers

971 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 ...

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167 views

### reference-request for connection between numerator of zeta-function and characteristic polynomial of Frobenius on hyperelliptic curves over finite fields

Let $H$ be a hyperellipitic curve of genus $g$ defined over
$\mathbb{F}_q$. The Frobenius endomorphism operates on the divisor class
group of $H$ and satisfies a characteristic polynomial ...

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votes

**1**answer

549 views

### Multiplication of matrices in GF(2) and R

$H$ is an $n \times n$ matrix with elements in $ \{ -1,1 \}$
$G$ is an $n \times k$ matrix with elements in $GF(2)$ and also upper triangular, invertable
$m$ is an $k \times 1$ vector with elements ...

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votes

**2**answers

173 views

### Cases of almost-linear time solvable linear systems

Let a square $n\times n$ real matrix ${\bf A}$ and two vectors ${\bf x}$ and ${\bf b}$ of length $n$, such that $${\bf A}{\bf x}={\bf b}.$$
Solving for ${\bf x}$ through standard Gaussian Elimination ...

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**1**answer

212 views

### Low degree approximation of the polynomial extension of the logical-or function

Let $x\in\{0,1\}^n$ be a binary vector of dimension $n$, and let $OR(x)$ be the "logical or" function (i.e., returns $1$ if at least one of the coordinates is $1$ and otherwise returns $0$).
Consider ...

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**1**answer

196 views

### Finding an embedding efficiently in field extension of finite field

We know that $GF(p^c)$ is a subfield of $GF(p^{cn})$. Also we know that elements in $GF(p^c)$ can be represent by degree $c$ polynomials with coefficients in $\mathbb Z_p$, where multiplication is ...

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votes

**1**answer

598 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, ...

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275 views

### Minimum Growth Rate of Hamming Weight of Multiples of Primitive Polynomials

Let $F_2[x]$ denote the ring of polynomials over the field of 2 elements.
Richard Brent has a page on finding primitive trinomials in $F_2[x]$ of huge degree at ...

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**2**answers

474 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 ...

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**3**answers

629 views

### Field with cyclic product group

If a field has a cyclic multiplicative group, is it necessarily finite?

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**3**answers

819 views

### Fast Vandermonde matrix multiplication over finite field

Let $V_{i,j}=x_i^j$ where $x_i\in\mathbb F_q$ for $1\le i\le n,1\le j\le n$ be a Vandermonde matrix over finite field $\mathbb F_q$.
I wish to know the currently known fastest algorithms for ...

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317 views

### non-split Frobenius map

there are 3 simple groups arising from $SO(8)$, i.e.$ SO(8,F_{q})$, $2D_4(q,q^2)$,$3D_4(q,q^3),$
so I want to know
For the $3D_4(q,q^3),$, what is the corresponding Frobenius?

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238 views

### Sums of three squares in $GF(q)[t]$

Let $q$ be a power of an odd prime number $p>3,$ say $q=p^r$ with $r \geq 1.$
It is known that every polynomial $M$ in $GF(q)[t]$ is a sum of at most $3$ squares
with some conditions on degrees. ...

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votes

**1**answer

268 views

### A kind of `pell` equation in characteristic $2.$

Given polynomials $a(t),b(t) \in K[t]$ where $K$ is a finite extension of $GF(2)$,
with
$$
a(t) \neq 0,
$$
we consider the equation
$$
x^2+axy+by^2=1
$$
with unknowns $x,y$ also polynomials in ...

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votes

**2**answers

523 views

### Reducible trinomials $x^{odd}+x^2+s(t)$ in characteristic $2$

Let $F$ be a finite field of characteristic $2$. Let $m \geq 2 $ be a positive integer.
Seems unknown
if the trinomial
$$
T(t,x) = x^{2m+1}+x^2+s(t) \in F[t][x]
$$
(more explicitly, the ...

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votes

**2**answers

397 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 ...

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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 ...

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vote

**2**answers

649 views

### best deterministic complexity for factoring polynomials over finite field

I would like to know currently what's the best deterministic complexity for factoring polynomials over finite field (without the assumption of GRH)? I have searched on google, there are many source, ...

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**2**answers

849 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 field ...

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**7**answers

588 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 ...

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**1**answer

792 views

### nontrivial cube root of unity [closed]

Hi,
I have a finite field Fp with p = 11 mod(12) and I am trying to get the third nontrivial root of unity in Fp^2 = Fp^2[x]/(x^2+1). So, i need x where x^3=1.
Somehow I came into a source saying ...

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votes

**1**answer

666 views

### Elliptic curve over finite field: scalar multiplication

I'm implementing arithmetics for elliptic curves over secp256r1 as a homework assignment.
For scalar multiplication, the assignment specifically specifies that $k$ is "any hexidecimal encoded ...

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votes

**1**answer

3k views

### Number of n-th roots of unity over finite fields [closed]

How many $n$-th roots of unity does a finite field $\mathbb{GF}\left(p^k\right)$ have? ($p$ is prime). And then kind of related to that, is there a finite field with exactly $n_1,\ldots,n_N$ ...

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votes

**1**answer

1k views

### Irreducibility of some trinomials modulo $p$

Let $n>1$ be an integer. An old result of Selmer,
See Theorem 1, page 289 in
http://www.mscand.dk/article.php?id=1472,
(If the link does not work try googling: ...

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182 views

### Binary `neighbors` perfect polynomials, instead of `consecutive`

Inspired by Andrei's nice solution of 52609, (namely consecutive perfect polynomials):
Denote by $A$ the full ring of polynomials in one variable $t$ over the ...

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votes

**1**answer

267 views

### Can two Consecutive Polynomials both be perfect ?

Denote by $A$ the full ring of polynomials in one variable $t$ over the finite field with $q$ elements.
For any monic polynomial $P \in A$ define
$$
\sigma(P) = \sum_{d \mid P, d\, \text{monic}} d.
...

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330 views

### J. T. B. Beard's algorithm to catch `perfect polynomials`

For any monic polynomial $P$ with coefficients in a finite field of order $q$, we put
$$
\sigma(P) = \sum_{d \mid P, d \text{ monic}} d.
$$
Observe that $P$ and $\sigma(P)$ have the same degree.
...

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votes

**1**answer

902 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 ...

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votes

**1**answer

234 views

### Quadratic forms without common zeroes

A question which, I suppose, is as easy as abc for an expert. For a given finite field $F$ of odd characteristic (if needed, the characteristic is $3$ and the size of $F$ is large), do there exist two ...

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133 views

### elementary question on ECDLP

If $\mathbb{F}_q$ is a finite field and the elliptic curve E is defined over finite field $\mathbb{F}_p$ such that ECDLP is hard in E($\mathbb{F}_p$), where $q, p$ are prime and $q \ll p$. Let T $\in ...

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328 views

### Finch's sequence over $\mathbb{F}_3$

In http://algo.inria.fr/csolve/seqmod3.pdf -- "Periodicity in sequences mod 3" Steven Finch (also cited in Sloane's OEIS A112683) defines the following sequences in $\mathbb{F}_3$:
For each positive ...

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votes

**2**answers

1k 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. ...