Absorbent property, Krasner type lemmas and spectral norms for a class of valued fields

作者：Popescu Sever Angel^*

来源：Proceedings of the Japan Academy Series A-Mathematical Sciences, 2013, 89(10): 138-143.

DOI：10.3792/pjaa.89.138

• Valued fields
• Krasner Lemma
• spectral norms

Let (K, phi) be a perfect valued field of rank 1, let (phi) over bar be an extension of the absolute (multiplicative) value phi to a fixed algebraic closure (K) over bar and let parallel to center dot parallel to(phi) be the corresponding spectral norm on K. Let ((K) over bar, parallel to center dot parallel to(phi)) be a fixed completion of ((K) over bar, parallel to center dot parallel to(phi)). In this paper we generalize a result of A. Ostrowski [8] relative to the absorbent property of a subfield, from the case of a complete non-Archimedian valued field of characteristic 0 to our ring ((K) over bar, parallel to center dot parallel to(similar to)(phi)) (see Theorem 1, Theorem 4). We also apply these results to discuss in a more general context the following conjecture due to A. Zaharescu (2009): (For any x, y is an element of C-p-the complex p-adic field, there exists t is an element of Q(p)-the p-adic number field, such that Q(p)((x, y) over tilde) = Q(p)((x + ty) over tilde), where (L) over tilde means the p-adic topological closure of a subfield L of C-p in C-p).

• 出版日期2013-12