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The minimum discriminant of totally real octic fields. (English) Zbl 0719.11079
Summary: The minimum discriminant of totally real octic algebraic number fields is determined. It is 282,300,416 and belongs to the ray class field over $${\mathbb{Q}}(\sqrt{2})$$ of conductor $$(7+2\sqrt{2}):{\mathcal F}={\mathbb{Q}}(\sqrt{\alpha})$$ for $$\alpha =(7+2\sqrt{2}+(1+\sqrt{2})\sqrt{7+2\sqrt{2}})/2$$. There is - up to isomorphy only one field of that discriminant. The next two smallest discriminant values are 309,593,125 and 324,000,000. For each field we present a full system of fundamental units and its class number.

##### MSC:
 11R80 Totally real fields 11R29 Class numbers, class groups, discriminants 11R27 Units and factorization
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