Monogenic dihedral quartic extensions
A number field whose ring of integers has a power integral basis of the form \(O_K= \mathbb{Z}[\vartheta]\), is called a monogenic number field. It is expected that monogenic number fields of a high degree, are very seldom. Define \(L^{(m)}(X)=\{\text{set}\) of monogenic fields of absolute discriminants \(\le X\) with Galois group \(S_n\) for its Galois closures\}. Much has been investigated regarding \(|L^{(m)}(X)|\). Bounds have been established for some explicit situations. As to monogenic \(D_4\)-quartic fields \(X\) (\(D_4\cong\) dihedral group of order 8), the main result of the paper is: Theorem 1.1 \(|L^{(m)}(X)|= O(X^{\frac{3}{4}}\cdot (\log X)^3)\). There are in this paper some other, be it somehow technical, results for such \(L^{(i)}(X)\) for some particular kinds of \(X\). It is interesting to learn some techniques in doing calculations in this paper.
scientific article; zbMATH DE number 7140973
scientific article