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CHARACTERISATION OF PRIMES DIVIDING THE INDEX OF A CLASS OF POLYNOMIALS AND ITS APPLICATIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 3 / December 2024
- Published online by Cambridge University Press:
- 01 April 2024, pp. 460-467
- Print publication:
- December 2024
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- Article
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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.
