The monogenicity and Galois groups of certain reciprocal quintinomials
The monogenicity and Galois groups of certain reciprocal quintinomials
We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for ${\mathbb Z}_K$, the ring of integers of $K={\mathbb Q}(\theta)$, where $f(\theta)=0$. For $n\ge 2$, we define the reciprocal quintinomial \[{\mathcal F}_{n,A,B}(x):=x^{2^n}+Ax^{3\cdot 2^{n-2}}+Bx^{2^{n-1}}+Ax^{2^{n-2}}+1\in …