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ON A CONJECTURE OF LENNY JONES ABOUT CERTAIN MONOGENIC POLYNOMIALS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 1 / August 2024
- Published online by Cambridge University Press:
- 21 November 2023, pp. 72-76
- Print publication:
- August 2024
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Let
$K={\mathbb {Q}}(\theta )$ be an algebraic number field with 
$\theta $ satisfying a monic irreducible polynomial 
$f(x)$ of degree n over 
${\mathbb {Q}}.$ The polynomial 
$f(x)$ is said to be monogenic if 
$\{1,\theta ,\ldots ,\theta ^{n-1}\}$ is an integral basis of K. Deciding whether or not a monic irreducible polynomial is monogenic is an important problem in algebraic number theory. In an attempt to answer this problem for a certain family of polynomials, Jones [‘A brief note on some infinite families of monogenic polynomials’, Bull. Aust. Math. Soc. 100 (2019), 239–244] conjectured that if 
$n\ge 3$, 
$1\le m\le n-1$, 
$\gcd (n,mB)=1$ and A is a prime number, then the polynomial 
$x^n+A (Bx+1)^m\in {\mathbb {Z}}[x]$ is monogenic if and only if 
$n^n+(-1)^{n+m}B^n(n-m)^{n-m}m^mA$ is square-free. We prove that this conjecture is true.
