Dihedral extensions of degree 8 over the rational \(p\)-adic fields
The author exhibits all extensions \(L/ \mathbb{Q}_p\) with Galois group isomorphic to the dihedral group \(D_4\) of order 8 and shows that there are exactly 0, 1, and 18 such extensions \(L\) respectively for \(p \equiv 1 \pmod 4\), \(p \equiv 3 \pmod 4\), and \(p = 2\). Examples are \(\mathbb{Q}_p (\sqrt {-1}, \root 4 \of {p})\), \(\mathbb{Q}_2 (\sqrt {1 + \sqrt 2}\), \(\sqrt {-1})\), \(\mathbb{Q}_2 (\root 4 \of {2}, \sqrt {-1})\), and \(\mathbb{Q}_2 (\sqrt {1 + \sqrt {-2}}, \sqrt {-5})\), where \(\mathbb{Q}_p\) denotes the rational \(p\)-adic field and \(p\) is a prime number. A similar problem for the quaternion group of order 8 was solved by \textit{G. Fujisaki} in [Proc. Japan Acad., Ser. A 66, No. 8, 257-259 (1990; Zbl 0732.11065)].
scientific article; zbMATH DE number 799712