# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 = ...
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### fields and subspaces

Let F be a field and let V=F^F, which is a vector space over F. Let w be the set of all functions f element of V satisfying f(1)=f(-1). Is W a subspace of V? a. Has the zero vector b. closed under ...
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### “$q$-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 ...
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### 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 ...
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### permutation polynomial

If we have GF(4) as an extension field, we can define a permutation polynomial of GF(4) like L(x), a linearized polynomial, of the followinf form: L(x)= \sum_{s=0}^{\r-1} a_s x^(q^r)e Is it possible ...
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### Solutions of $x^d=1$ in a finite field

Let's consider the polynomial $x^d-1$. Theory tells us that it can have at most $d$ roots in (any extension of) a given field. Here's my problem: let $A$ be the vector space spanned by ...
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### Can a set containing a single vector from a vector space over a finite field be linearly dependent?

Take the set $S=\{v=(1,1)\}\subset F_2 ^2$. $v+v=(0,0)$ is a linear combination of vectors from $S$. Is $S$ linearly dependent?
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### generating set for finite fields

let us consider GF(2^n) as a vector space over GF(2), Is it possible to find a generating set for GF(2^n)? How can ew find it? I want to define a linear transformation of GF(2^n) to itsefl.
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### elementary abelian groups and finite fields

in group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order $p$, where $p$ is a prime; it is a particular kind of $p$-group. now suppose that we ...
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### automorphism of fields

let we consider $GF(p^n)$ as a vector space over $GF(p)$, $p$ is prime. Also we want to have an invertible linear map on $GF(p^n)$, (automorphism of $GF(p^n)$). on the other hand we know that A field ...
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### Is vectorspace trivial under these conditions?

Let $R$ be a ring. Looking for a left $R$-module free over abelian group $A$, I arrived at $\left|R\right|\otimes A$ with $r.\left(s\otimes a\right)=rs\otimes a$ where $\left|R\right|$ denotes the ...