# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 $T =\begin{bmatrix}3& -1 &1\\-1 & 5 & -1\\ 1 & -1 & 3\end{bmatrix}$ ...
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### 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 $ker(f)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### $GL_n(\mathbb F_q)$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### Permutation matrix to “invert” part of a matrix?

I am working with a Linear Code, C, over $F_{2^5}$, and its dual, $C^{\perp}$. I have the generator matrix for $C$, $G$, and have calculated the generator matrix for $C^{\perp}$, $G^{\perp}$. I need ...
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### clarification on Taylor's Formula

In Linear Algrebra form Hoffman and Kunze, the Taylor's Formula is stated as follows: Theorem 5. (Taylor's Formula) (page 129) Let $\mathbb{F}$ be a field of characteristic zero, $c\in \mathbb{F}$, ...
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### Is there any field of characteristic 4 or any other composite number? [duplicate]

Possible Duplicate: Characteristic of a field is $0$ or prime Is there any field of characteristic 4? Or any other composite number?
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### On the order of elements of $GL(2,q)$?

There's a particular property of the elements of $GL(2,q)$, the general linear group of $n\times n$ matrices over a finite field of order $q$, that I don't understand. First, I know that the order of ...
### Some iterate of a linear operator over $\mathbf F_q$ is a projection
If $T$ is an endomorphism of a finite-dimensional vector space $V$ over a finite field, then how can I show that there exists a positive integer $r$ such that $T^r$ is a projection operator?
I want to know how many $2\times 2$ orthogonal matrices exist over the ring $\mathbb{Z}_n$ or the field $\mathbf{F}_p$. And how many $2\times 1$ orthogonal vectors exist over the ring $\mathbb{Z}_n$ ...