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On an Existence Lemma in Valuation Theory

Published online by Cambridge University Press:  22 January 2016

Masayoshi Nagata ,
Tadasi Nakayama  and
Tosiro Tuzuku
Masayoshi Nagata
Affiliation:
Mathematical Institute, Nagoya University
Tadasi Nakayama
Affiliation:
Mathematical Institute, Nagoya University
Tosiro Tuzuku
Affiliation:
Mathematical Institute, Nagoya University
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Recently one of the writers used, in proving a theorem on the commutativity of certain division rings, the following lemma :

I. Let L be a field and K be its proper subfield. Except either when L is of characteristic p ≠ O and absolutely algebraic or ivhen L is algebraic and purely inseparable over K, there exists a pair of distinct (special exponential) valuations in L which coincide on K.

II. Let K be a field which is either of characteristic 0 or not absolutely algebraic, and L be its separable finite extension. There exist then infinitely many valutions in L which are of 1st degree over K.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 6 , October 1953 , pp. 59 - 61
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1953

References

1) Nakayama, T., “On the commutativity of certain division rings,” forthcoming in Canad. J. Math. Google Scholar

2) On considering the Galois field of L over K instead of L, it is then easy to show that there are infinitely many valuations in K each of which has (L : K) distinct prolongations in L.

3) Moriya, M., “Rein arithmetischer Beweis über die Unendlichkeit der Primideale 1. Gradesnus einem algebraeschen Zahlkõrper,” Jour. Fac. Sci. Hokkaido Univ. ser. I.vol. 9(1930)Google Scholar.

4) First the second writer gave a somewhat clumsy proof to I, for the purpose of using in his note 1), and then the first and “the third writers gave simpler proofs, which the three writers cooperated in further simplifying and refining into the present one.

5) We could without loss in generality start with an a contained in I, and then wording would be simplified a little in the sequel.

6) We choose any one of the allowed ones.