Let p be a prime number. Let $n\geq 2$ be an integer given by $n = p^{m_1} + p^{m_2} + \cdots + p^{m_r}$, where $0\leq m_1 < m_2 < \cdots < m_r$ are integers. Let $a_0, a_1, \ldots , a_{n-1}$ be integers not divisible by p. Let $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta \in {\mathbb C}$ a root of an irreducible polynomial $f(x) = \sum _{i=0}^{n-1}a_i{x^i}/{i!} + {x^n}/{n!}$ over the field $\mathbb Q$ of rationals. We prove that p divides the common index divisor of K if and only if $r>p$. In particular, if $r>p$, then K is always nonmonogenic. As an application, we show that if $n \geq 3$ is an odd integer such that $n-1\neq 2^s$ for $s\in {\mathbb Z}$ and K is a number field generated by a root of a truncated exponential Taylor polynomial of degree n, then K is always nonmonogenic.

Let q be a prime number and $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta $ a root of an irreducible trinomial $x^{6}+ax+b$ having integer coefficients. In this paper, we provide some explicit conditions on $a, b$ for which K is not monogenic. As an application, in a special case when $a =0$ , K is not monogenic if $b\equiv 7 \mod 8$ or $b\equiv 8 \mod 9$ . As an example, we also give a nonmonogenic class of number fields defined by irreducible sextic trinomials.