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On regular number fields. (Sur les corps de nombres réguliers.) (French) Zbl 0704.11040
Fix a prime number $$\ell$$ and let $$\zeta$$ denote a primitive $$\ell$$-th root of unity. Let $$K$$ be a number field with divisor group $$D_ K$$ and write $$\mathcal D_ K$$ for the multiplicative tensor product $$\mathbb Z_{\ell}\otimes_{\mathbb Z}D_ K$$. One can define a surjective map (called Gras’s logarithm), $$\text{lg}: \mathcal D_ K\to \text{Gal}(Z/K)$$, where $$Z$$ is the composite field of the $$\mathbb Z_{\ell}$$-extensions of $$K$$. For a finite or a real place $$\wp$$ of $$K$$, $$\text{lg}(\wp)=1$$ whenever $$\wp$$ is real or lies over $$\ell$$, otherwise $$\text{lg}(\wp)$$ is a topological generator of the decomposition group $$D_{\wp}(Z/K)\simeq\mathbb Z_{\ell}$$ associated to $$\wp$$ in the abelian extension $$Z/K$$. A finite set $$S$$ of places of $$K$$ is called primitive when the $$\text{lg}(s)$$, $$s\in S$$, form a $$\mathbb Z_{\ell}$$-basis of a pure submodule of $$\text{Gal}(Z/K)$$. An $$\ell$$-extension $$L/K$$ is called primitively ramified if the set $$S$$ of places of $$K$$ that ramify tamely in $$L/K$$, is primitive. $$K$$ is called regular (with respect to $$\ell$$) if the $$\ell$$-Sylow subgroup $$R_ 2(K)$$ of the kernel in $$K_ 2(K)$$ of the regular symbols attached to the non-complex places of $$K$$, is trivial. When $$K$$ contains the maximal real subfield $$k=\mathbb Q(\zeta +\zeta^{-1})$$ of the cyclotomic field $$\mathbb Q(\zeta)$$ several equivalent characterizations of regularity can be given, one of which says that $$K$$ is regular if and only if $$K$$ verifies Leopoldt’s conjecture (with respect to $$\ell)$$ and the torsion submodule $${\mathcal T}_ K$$ of $$\text{Gal}(M/K)$$, with $$M$$ the maximal $$\ell$$-ramified, abelian $$\ell$$-extension of $$K$$ which decomposes completely at the infinite places, is zero. A classical example of a regular number field is provided by the cyclotomic field $$\mathbb Q(\zeta)$$ if and only if $$\ell$$ is a regular prime in the usual terminology. Writing $$\delta_ K$$ for the defect of Leopoldt’s conjecture for $$K$$, $$K$$ is called an $$\ell$$-rational field if $${\mathcal T}_ K=\{0\}$$ and $$\delta_ K=0$$. The main results of the paper can now be formulated:
1) Let $$K$$ contain the maximal real subfield $$k$$, and let $$L/K$$ be a Galois $$\ell$$-extension. Then the following conditions are equivalent: (i) $$L$$ is regular; (ii) $$K$$ is regular and $$L/K$$ is primitively ramified.
2) Let $$L/K$$ be a Galois $$\ell$$-extension. Then the following conditions are equivalent: (i) $$L$$ is $$\ell$$-rational; (ii) $$K$$ is $$\ell$$-rational and $$L/K$$ is primitively ramified.

##### MSC:
 11R29 Class numbers, class groups, discriminants 11R70 $$K$$-theory of global fields 11S15 Ramification and extension theory 19C99 Steinberg groups and $$K_2$$ 19F15 Symbols and arithmetic ($$K$$-theoretic aspects)
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