Infinite families of reciprocal monogenic polynomials and their Galois groups (Q2075861)

A monic irreducible polynomial \(f(x)\in \mathbb{Z}[x]\) is said to be reciprocal (resp. monogenic) if \(f(x)=x^{\deg (f)}f(1/x)\) (resp. if the ring of integers of the field \(\mathbb{Q}(\alpha ),\) where \(\alpha \) is a zero of \(f,\) is of the form \(\mathbb{Z+\alpha Z}+\cdot \cdot \cdot +\alpha ^{\deg (f)-1}\mathbb{Z}).\) In the paper under review, the author provides an irreducibility criterion for a certain class of polynomials with integers coefficients, and uses it to construct infinite families of reciprocal monogenic polynomials. More precisely, he proves that if \(a\geq 0,\) \(b\geq 1\) are integers, \(q\in \{3,5,7\}\) and \(r\geq 3\) is a prime primitive root modulo \(q^{2},\) then there exist infinitely many primes \(p\) such that a polynomial of the form \[ f_{(a,b,q,r,p)}(x):=\Phi _{2^{a}q^{b}}(x)+4q^{2}rpx^{\frac{\varphi (2^{a}q^{b})}{2}}, \] is monogenic (\(\Phi _{2^{a}q^{b}}\) is the cyclotomic polynomial of index \(2^{a}q^{b}\) and \(\varphi \) is the Euler function). As an application of this theorem, the author obtains from the class \(\{f_{(a,b,q,r,p)}(x)\}\) infinite families of reciprocal monogenic polynomials with prescribed Galois group. These results are non-trivial extensions of previous results of the author [Ramanujan J. 56, No. 3, 1099--1110 (2021; Zbl 1487.11095)] on reciprocal monogenic polynomials of degree \(6.\)