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CHARACTERISATION OF PRIMES DIVIDING THE INDEX OF A CLASS OF POLYNOMIALS AND ITS APPLICATIONS
Published online by Cambridge University Press: 01 April 2024
Abstract
Let 
${\mathbb {Z}}_{K}$ denote the ring of algebraic integers of an algebraic number field 
$K = {\mathbb Q}(\theta )$, where 
$\theta $ is a root of a monic irreducible polynomial 
$f(x) = x^n + a(bx+c)^m \in {\mathbb {Z}}[x]$, 
$1\leq m<n$. We say 
$f(x)$ is monogenic if 
$\{1, \theta , \ldots , \theta ^{n-1}\}$ is a basis for 
${\mathbb {Z}}_K$. We give necessary and sufficient conditions involving only 
$a, b, c, m, n$ for 
$f(x)$ to be monogenic. Moreover, we characterise all the primes dividing the index of the subgroup 
${\mathbb {Z}}[\theta ]$ in 
${\mathbb {Z}}_K$. As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group 
$S_n$, the symmetric group on n letters.
MSC classification
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 110 , Issue 3 , December 2024 , pp. 460 - 467
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
The author is thankful to IIT Madras for NFIG grant RF/22-23/1035/MA/NFIG/009034.
Dedicated to Professor Sudesh Kaur Khanduja
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