Let \(p\equiv -q \equiv 5\pmod 8\) be two prime integers. In this paper, we investigate the unit groups of the fields \( L_1 =\mathbb {Q}(\sqrt{2}, \sqrt{p}, \sqrt{q}, \sqrt{-1} )\) and \( L_1^+=\mathbb {Q}(\sqrt{2}, \sqrt{p}, \sqrt{q} )\). Furthermore , we give the second 2-class groups of the subextensions of \(L_1\) as well as the 2-class groups of the fields \( L_n =\mathbb {Q}( \sqrt{p}, \sqrt{q}, \zeta _{2^{n+2}} )\) and their maximal real subfields.

中文翻译：

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一些二次数域的单位群和二类群

令\(p\equiv -q \equiv 5\pmod 8\)为两个素数。在本文中，我们研究了域\( L_1 =\mathbb {Q}(\sqrt{2}, \sqrt{p}, \sqrt{q}, \sqrt{-1} )\)和\( L_1^+=\mathbb {Q}(\sqrt{2}, \sqrt{p}, \sqrt{q} )\)。此外，我们给出了\(L_1\)的子扩展的第二个 2 类群以及域\( L_n =\mathbb {Q}( \sqrt{p}, \sqrt{q }, \zeta _{2^{n+2}} )\)和它们的最大实子域。