## Discriminant minimal et petits discriminants des corps de nombres de degre 7 avec cinq places réelles. (Minimal discriminant and small discriminants of degree 7 number field with five real places).(French)Zbl 0653.12003

The author considers algebraic number fields k of degree $$7$$ with 5 real places, and gives a list of such fields with discriminant $$| d_ k| <5000000$$. He shows that this list contains all such fields with discriminant $$| d_ k| <3477288$$, from which it follows the minimal discriminant for these types of fields is -2306599.
With this result the problem of determining the minimal discriminants for algebraic number fields of degree 7 is completely solved [cf. the author, Ann. Inst. Fourier 34, No.3, 29-38 (1984; Zbl 0546.12004)]. - Moreover, the author conjectures that his list contains all such fields with discriminant $$| d_ k| <4467575$$.
Reviewer: H.Yokoi

### MSC:

 11R23 Iwasawa theory 11R21 Other number fields

Zbl 0546.12004
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