An infinite families of number fields with fixed indices arising from quintinomials of type \(x^n+ax^m+bx^2+cx+d\) (Q6635599)

The paper deals with powers of primes dividing the field index \(i(K)\) for nine infinite families of fields defined by a root of an irreducible polynomial appearing in the title. This is achieved with the use of Newton polygons in a modern version of Ore's theorem [\textit{Ö. Ore}, Math. Ann. 99, 84--117 (1928; JFM 54.0191.02); \textit{J. Guàrdia} et al., Trans. Am. Math. Soc. 364, No. 1, 361--416 (2012; Zbl 1252.11091)]. In particular, part (1) of Theorem 2.10 shows that if \(p\) is an odd prime and one has \(b\equiv-1\) mod \(p\), \(p\) divides \(a,b,c,d\), \(n=2+k(p-1)\) for some \(k\) not divisible by \(p\) and moreover \(\nu_p(d)>2\nu_p(c)\), then \(p\nmid i(K)\).