0
votes
3answers
29 views

Finding the order of elements in a Galois Field

Does there exist a Galois field GF(4)? GF(4)={0,1,2,3}; If we take this Galois field, then the element '2' is not having any degree..? So is it possible to construct GF(4) ?
0
votes
1answer
13 views

Prove the number of unordered pairs of linearly independent elements

Let $V$ be a vector space over $K$. Let $K={\mathbb{Z}}/{p\mathbb{Z}}$, and $\dim V=3$. We know that $V$ has $p^3$ elements. I need to show that the number of unordered pairs of linearly ...
1
vote
1answer
55 views

Sequences length for LFSR in the general case

An LFSR with a reducible polynomial can generate several sequences, depending on the initial value. My goal is to have an algorithm to compute those length without going through the enumeration of all ...
1
vote
1answer
38 views

Maps preserving roots of a polynomial function over finite fields

Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$. Let $S(P)=\{ ...
1
vote
1answer
49 views

Sequences length for LFSR when polynomial is reducible

An LFSR with polynomial 1+x4+x5 = (1+x+x2)(1+x+x3) can generate several sequences, depending on the initial value. If I did not made any mistake enumerating them, the sequences length are 3, 7 and 21. ...
2
votes
0answers
24 views

Blocking set for cosets of codimension $2$

In this paper following theorem is proved: If $V$ is vector space of dimension $n$ over a finite field $F$ of $q$ elements then any subset of $V$ which meets every hyperplane of $V$ contains at least ...
3
votes
0answers
124 views

Find minimal set of cosets $C$, so that each $2$ vectors in $A_n$ are in one coset in $C$

Let $F_2^n$ be the set of all vectors of length $n$ with values of $0$ or $1$ and $A_n$ = $F_2^n \setminus(11\ldots1)$. Set $A_n$ contains all vectors except one with all $1$s. We can consider cosets ...
1
vote
1answer
20 views

Dimension reduction over finite fields

Let $\mathbf b_1, \cdots, \mathbf b_n$ be a basis for $\mathbb F^n_q$ (where $\mathbb F_q$ is a finite field of size $q$). Assume $c \in \mathbb F_q$ is a uniformly randomly chosen number. For a given ...
2
votes
0answers
11 views

Subspace-evasive set performance in the random case

A subspace evasive set is defined as a large subset of a vector space which has small intersection with any $k$ dimensional affine space. That is, it "evades" all affine subspaces of small enough ...
1
vote
2answers
49 views

Weird field notation

I have a question: Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define $$S = \{x\in\mathbb{F}^5 \mid x_i = ...
2
votes
1answer
86 views

What is the number of distinct subgroups of the automorphism group of $\mathbf{F}_{3^{100}}$?

Let $G$ denote the group of all the automorphisms of the field $\mathbf{F}_{3^{100}}$ that consists of $3^{100}$ elements. What is the number of distinct subgroups of $G$?
2
votes
3answers
40 views

Some properties of elements of the finite field GF(q)

Consider the finite field $GF(q)$ and an element $w$ of order $n$ which is an $n$th root of unity in this field. Could someone give me explanation or reference to the following questions regarding ...
6
votes
1answer
84 views

Equation over $\mathbb F_2$

Given equation over $\mathbb F_2$: $$x_1x_2x_3+x_4x_5x_6=0$$ It has $50$ solutions. Let $N$ be set of solutions. If we give some linear dependencies to variables we will get cosets (linear subspace + ...
1
vote
1answer
66 views

Cyclotomic Cosets and Minimal Polynomial for 45

Currently I am working on matlab in order to find Cyclotomic Cosets for 45. As 45 in not in the format of 2^m-1, matlab give me an error. I am trying to write algorithm in matlab/octave for my ...
1
vote
1answer
31 views

How many different bases in $\mathbb{Z}/p\mathbb{Z}$

Let $K = \mathbb{Z}_p$, for some prime $p$, and $\text{dim}\:V = n$. $V$ is a vector space over $K$. I need to find out how many different bases are in $V$. Now I know the answer is the product of ...
0
votes
1answer
25 views

matrix representation of Frobenius map

I am in an urgent need to know what is the matrix representation of Frobenius map for finie field like GF(4). Let suppose the basis of GF(4) be {1, a+1}. We know that the frobenius map is generator of ...
1
vote
1answer
29 views

“$q$-linear envelopes” of $\mathbb{F}_p$-subspaces

Let $V$ be a vector space over an algebraically closed field $k$ of characteristic $p>0$, and denote by $V_q$ the vector space obtained from $V$ by restricting scalars to $\mathbb{F}_q$, where ...
2
votes
1answer
75 views

Having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space over $\mathbb{F}_p$.

As the title states, I'm having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space V over $\mathbb{F}_p$. Clearly it is dimension $n$. How can we work with this vector space? ...
1
vote
1answer
21 views

matrix for frobenius map of finite fields

I would be thankful if you could help me : I have studied many things about Galois fields, but now I am not sure about my understanding of frobenius maps. For example can anyone help me the matirx of ...
2
votes
1answer
37 views

Non-obvious deduction regarding conjugates in $\text{GL}_2(\mathbb{F}_p)$

Let $\text{GL}_2(\mathbb{F}_p)$ act on $\mathbb{F}_p^2,$ the set of $2$-vectors with entries in $\mathbb{F}_p$, by matrix multiplication. $[\,$Prove that$\,]$ for any ...
2
votes
1answer
63 views

Structure of a module and finite fields

I am trying the following problem which appeared on an exam for a grad course: Determine the structure of finite field $F$ as a module over $F_p[x]$ when $xv=Lv$. Here $L(z)=z^p$ and ...
1
vote
0answers
39 views

About Degree of Polynomial

Consider a set $M_{m\times m}(F)$ of matrices of order $m\times n$ over the finite field $\mathbb{F_p}$. The set $M_{m\times m}(\mathbb{F_p})$ forms a $Ring$ under the binary operations (addition, ...
1
vote
1answer
29 views

Creating a matrix such that all the sub-matrices are max rank

Let $A\odot B$ denote the elementwise multiplication of matrices $A$ and $B$. Given a binary matrix $B_{m \times n}=[b_{ij}]$, $b_{ij} \in \{0,1\}$, I want to find a matrix $A=[a_{ij}]$, $a_{ij}\in ...
1
vote
1answer
48 views

Basis of finite field as vector space

If we consider GF(8) as a vector space over GF(2), what are the basis for GF(8)? and How can we define a dual space for GF(8) as a vector space?
0
votes
0answers
45 views

Connecting inner products and the trace of a matrix

Hello, I'm currently trying to work through this question. However, I am struggling with understanding what I should do. I am aware of all of the definitions used throughout, however I cannot link ...
0
votes
1answer
52 views

Linear independence of finite field elements and subfields

Let $q$ be a prime power and $n=lm$ an integer with $l,m>1$. We know that the finite field $GF(q^n)$ is a $n$-th dimensional vector space over $GF(q)$, and it is also a $l$-th dimensional vector ...
1
vote
0answers
61 views

Extension field of F2 , expressing roots and primitive elements in that field

Let $\Phi$ be an extension field of $\Bbb{F}_2$ of extension degree s >1. Let $a(x)$ be a non-zero polynomial with the coefficients in $\Bbb{F}_2$. (a) Show that if $\beta$ is a root of the ...
0
votes
1answer
43 views

find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
1
vote
3answers
33 views

Linear Independence for different fields

I have a statement for a space over $R^n$: {x, y, z} is linearly ind. $\implies$ {x + y, x + z, y + z} is linearly independent Quick proof: a(x+y) + b(x+z) + c(y+z) = 0 $\implies$ (a+b)x + ...
0
votes
1answer
47 views

basis for finite fields as a vector space

If we consider GF(4) as a vector space over GF(2), the basis of GF(4) includes two elements 1 and a. Due to this fact that for an arbitrary vector space we can find several basis, what are other bases ...
3
votes
0answers
30 views

Action of $\mathbb{F}_{p^2}^\times/\mathbb{F}_{p}^\times$ on $P^1(\mathbb{F}_p)$

Let $p$ be prime. Let $\alpha$ be a generator of the finite field $\mathbb{F}_{p^2}$. So, $\mathbb{F}_{p^2}=\mathbb{F}_p[\alpha]$. Multiplication by $\alpha$ is an $\mathbb{F}_p$-linear operator on ...
1
vote
0answers
19 views

Find maximum rank vector

I have a $n\times n$ matrix with coefficients from a finite field $F$. For any vector $v$ of $F^n$, I consider the sequence: $v_1=v$ and $v_{n+1}=M.v_n$ The rank of $v$ is the rank of the sequence ...
2
votes
2answers
49 views

Finding the kernel and image of a linear transformation over the field $\Bbb Z_2$

Given the A vector space $V$ over the field $\Bbb Z_2$ and a linear map $t:V \to V $ .following matrix $T =\begin{bmatrix}3& -1 &1\\-1 & 5 & -1\\ 1 & -1 & 3\end{bmatrix}$ ...
1
vote
1answer
46 views

Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
2
votes
1answer
138 views

Eigenvalue problems for matrices over finite fields

Suppose I have a symmetric matrix A with entries in a finite field. In particular, I have the case in mind where $A \in \{0,1\}^{n \times n}$ and want to treat the entries as elements of $GF(2)$. How ...
0
votes
0answers
22 views

matirx of Frobenius map

Galois group of GF(8) is a cyclic group of order 3 . How is the matrix representation of generator of that? It is clear that its generator is Frobenius map? I think it is 3*3 matrices , one of them ...
0
votes
1answer
63 views

System of linear equations over Finite Field with restriction on variables

Given two vectors $x$ and $y$, where each element $x_i, y_i$ is from a finite field. I have the restriction that for each of these variables only about half of the elements of this finite field are ...
0
votes
0answers
57 views

automorphism group

an endomorphism of a vector space V is a linear operator V → V. and an automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is ...
1
vote
2answers
94 views

Finite fields, Linear Algebra

I just started my first upper level undergrad course, and as we were being taught vector spaces over fields we quickly went over fields. What confused me was when the book (Advanced Linear Algebra, ...
5
votes
1answer
95 views

$GL_n(\mathbb F_q)$ has an element of order $q^n-1$

For fixed prime power $q$ show that the general linear group $GL_n(\mathbb F_q)$ of invertible matrices with entries in the finite field $\mathbb F_q$ has an element of order $q^n-1$. I tried to ...
1
vote
1answer
50 views

most efficient way to find distinct complimenting subspaces over a finite field

Let $V$ be a $n$-dimensional vector spaoce over $\mathbb{F}_p$ and let $W$ be a $k$-dimensional subspace. What's the most efficient way to algorithmically write down a basis for each distinct ...
0
votes
1answer
93 views

Count the number of rational canonical form&find similarity classess

For a finite field $F=F_q$ having $q=p^d$ elements ($p$ a prime integer), compute the number of similarity classes in the vector space $M_n(F)$ of $n\times n$ matrices over $F$. (Maybe count the ...
2
votes
2answers
58 views

Linear independence over $\mathbb{F}_p$ for varying primes

Let $v_1,\ldots,v_k\in \mathbb{F}_p^n$ be a set of vectors, where $p$ is a prime. Assume further that the components of each vector can be represented by integers smaller than some integer $k$. Is ...
1
vote
1answer
174 views

How to find null space of a matrix over $\mathbb{GF}(2)$?

How to find null space of a matrix over $\mathbb{GF}(2)$ ? Are there any algorithms available? I am not sure here is appropriate to ask this or not: Is there any routines for MATLAB or Maple?
1
vote
1answer
55 views

Factoring a polynomial over finite field $\,F_3$ that has a root

A question I am struggling with. We are asked to factor $\,f(x)=x^2+x+1$ over the field $F_3 =\{0,1,2\}$ So, I checked for a root, and I saw that $f(1) = 1^2+1+1 =0$ (because $3=0$ in $F_3$) that ...
3
votes
0answers
459 views

How can I solve system of linear equations over finite fields in WolframAlpha?

Is it possible to solve system of linear equations over finite fields using Wolfram Alpha? If yes, how can I do that? Let us take a system $x+y+z=0$, $2x+y+2z=0$, $x+3y+z=0$. If I want to solve this ...
2
votes
1answer
56 views

A problem about class equation

Let $k$ be a finite field, where $|k|=q$ and $\operatorname{char}k\neq2$, and let $$D=\{A\in\operatorname{SL}(2,k)\mid A \text{ is diagonalizable}\}.$$ Prove that $$|D|=2+\tfrac{1}{2}\cdot(q+1)\cdot ...
1
vote
0answers
93 views

Number of ways to decompose the space $\mathbb F^n_2$ into a direct sum of two spaces

How many ways can $\mathbb F^n_2$ be decomposed into a direct sum of two subspaces? Basically how do I find the number of decompositions $\mathbb F^n_2 = \mathbb F^k_2 \bigoplus \mathbb F^{n-k}_2$ ...
4
votes
1answer
445 views

Find the number of n by n matrices conjugate to a diagonal matrix

(a) Find the number of matrices of size $n$ by $n$ over the field of two elements which are conjugate to a diagonal matrix. What is the answer for $n = 4$? (b) What is the number of $n$ by $n$ ...
0
votes
0answers
116 views

Solve system of matrix equations in finite field

I have the following system of matrix equations: \[ X_1 = A_1 X_2 B_1, \] \[ X_2 = A_2 X_1 B_2; \] where $A_i$, $B_i$ and $X_i$ are $n\times n$ matrices (for even $n$) over a finite field ...