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Results tagged with finite-fields
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user 3208
A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.
1
vote
0
answers
570
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Eigenvalue of a linear map over finite field
Let $ F_q $ be a finite field with $ q $ elements.
Let $ g $ be a multiplicative generator of $ F_{q^2}^* $.
It implies that
$ <g^{q+1}> = F_q^* $.
Let $ l $ be a prime greater than $ q^2-1 $ and …
- 503
0
votes
0
answers
101
views
Eigenvalues of the Cayley-like graph
Let $ F_q $ be a finite field of characteristic 2.
Let $ x^2 + Sx +P \in F_q[x] $ be an irreducible polynomial over $ F_q $,
and let $ g $ be one of its roots in $ F_{q^2} $.
Define a map $ M: F_{q^2 …
- 503
1
vote
0
answers
269
views
Number of solutions of a linear equation in a small subset.
Let $p$ be a prime. Let $F_p$ be the finite field of
$p$ elements. Let $A$ be a subset of $F_p$ of size $s$.
Assume that $s > 2$ is polylogarithmic in $p$.
Suppose that we want to count number o …
- 503
3
votes
1
answer
333
views
additive structure in a small multiplicative group of a finite field?
Let $p$ be a prime. Given a positive integer $n$, does there exist a
$\beta$ in an extension of $F_p$ such that
1) If $F_p[\beta] = F_{p^N}$, then $N > n^n$; ( $\beta$ lies in a high extension)
2) …
- 503
7
votes
2
answers
471
views
A quadratic form
Let $q$ be a power of 2. Let $P$ be the set of polynomials in
$F_q [x]$ of degree d or less.
Let $\mathbb{Z}$ be the ring of integers. For any $f \in P$, let $\psi(f)$ be the
number of distinct roo …
- 503