Monogenic Reciprocal Quartic Polynomials And Their Galois Groups

Suppose that $f(x)=x^4+A{x}^3+B{x}^2+Ax+1\in \mathbb{Z}[x]$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over $\mathbb{Q}$ and $\{1,\theta ,{\theta }^2,{\theta }^3\}$ is a basis for the ring of integers of $\mathbb{Q}(\theta )$, where $f(\theta )=0$. For each possible Galois group $G$ that can occur in the two cases of $A\mathrm{\ne }0$ with $B=0$, and $AB\mathrm{\ne }0$, we determine all monogenic polynomials $f(x)$ with Galois group $G$.