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ON THE DIMENSION OF PERMUTATION VECTOR SPACES
Part of:
Permutation groups
Finite fields and commutative rings (number-theoretic aspects)
Basic linear algebra
Published online by Cambridge University Press: 03 April 2019
Abstract
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Let 
$K$ be a field that admits a cyclic Galois extension of degree 
$n\geq 2$. The symmetric group 
$S_{n}$ acts on 
$K^{n}$ by permutation of coordinates. Given a subgroup 
$G$ of 
$S_{n}$ and 
$u\in K^{n}$, let 
$V_{G}(u)$ be the 
$K$-vector space spanned by the orbit of 
$u$ under the action of 
$G$. In this paper we show that, for a special family of groups 
$G$ of affine type, the dimension of 
$V_{G}(u)$ can be computed via the greatest common divisor of certain polynomials in 
$K[x]$. We present some applications of our results to the cases 
$K=\mathbb{Q}$ and 
$K$ finite.
MSC classification
Primary:
11T06: Polynomials
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 100 , Issue 2 , October 2019 , pp. 256 - 267
- Copyright
- © 2019 Australian Mathematical Publishing Association Inc.
Footnotes
The author was supported by FAPESP Brazil, grant no. 2018/03038-2.
References
Artin, E., Galois Theory, Notre Dame Mathematical Lectures, 2 (University of Notre Dame, Notre Dame, IN, 1942).Google Scholar
Lidl, R. and Niederreiter, H., Introduction to Finite Fields and Their Applications (Cambridge University Press, New York, 1986).Google Scholar
Pomerance, C., Thompson, L. and Weingartner, A., ‘On integers n for which x
n
- 1 has a divisor of every degree’, Acta Arith.
175 (2016), 225–243.Google Scholar
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