Non-normable extension of A p-adic field

Abstract

By a semi-topological field we understand a topological ring which is a field. A prime subfield of a complete locally bounded semitopological field of characteristic 0 is normable ([1], Theorem 3). In case an Archimedean topology is imposed on this prime subfield, the field itself is normable ([1], Corollary 7). However, the generalization which suggests itself of ShafarevichWs Theorem ([3], Theorem 1) on the normability of locally bicompact fields to the case of complete locally bounded ones turns out to be untrue in both remaining cases. In the case of a discrete topologization of the prime subfield, this follows from [2] and, in the present paper, this will be proven for the p-adic topologization of the prime subfield.Let p and q be prime numbers. We denote by R the field of rational numbers, we denote by R n, where qn n is a natural number, the field R(~rp), and U R n = S. The completion of S in the p-adic norm we denote by T. The elements of S of the form pmi/qni, where miandn i are natural numbers, we call monomials.Let x be an arbitrary element of T. We represent x in the form x = lira ~i, where ~i E S, i.e., N(xi+l-xi) -*+ % where N is the logarithmic p-acid norm. Each ~i is representable in the form of a sum of different monomials with natural coefficients not exceeding p. As i-* + oo, the coefficients of all the xj are stabilized, starting with some ordinal position and, letting aj stand for the limiting coefficient of xj, we consider ~jEj ajxj, where J is the set of all subscripts of the xj for all the ~i, excluding those j for which aj --0. Then, (*) for each real n there exists no more than a finite set of jEJ such that N(xj) < n, since, starting with that j for which N(ok+ 1 -ok) -n, these xj and their coefficients do not change. That condition (*) holds is obviously equivalent to the correctness of the definition of ~jEJajxj. For these same reasons, N(cri-~jEjajxj) ~+ % i.e., x = lim cr i = NjEJ ajxj. We shall prove finally that x is uniquely representable in such a form that there will be established a one-to-one correspondence between the elements of T and symbols of the form ~jEjajxj. If this were not the case, then 0 would have a non-trivial representation.By transferring the term with the least logarithmic norm to the left side (which exists, by virtue of condition (*)), we would obtain an equality between two expressions with unequal norms. If x ER n and xE-Rn+ 1, where x is a monomial, we then set e~(x) =-n. Thus, ~ is defined for all monomials, and defined uniquely. We set &(x) --N(x) + ~(x). We denote by ~ the set of those x= ~iEiaixi such that condition (*) holds for I with respect to ~. We define r =rain ~(xi) (the minimum exists, by iEI condition (*)). We set ~(0) =+ ~.Obviously ~I,(xy) -> ,P(x) + e~(y), and since the analogous assertion also holds for N(x), we then obtain for the monomials that ~(xy) >--r + ~(y). Let and let there be given some real n. We denote by I 0 and J0 respectively the sets of those i and j for which ~(xi) <n ~(y) and ~(yi ) ~ n-~(x). We denote by I* and ~F* r...

Locally unbounded topological fields with topological nilpotents

“…We have also shown in the previous lemma that, for each δ ∈ A and each U m ∈ B, there exists U s such that δU s ⊆ U m . This fact together with (8) shows that property (4) holds in E.…”

“…This is a missing example in the classification of complete first countable topological fields given by Mutylin [9, Table 1] (see also [13, p. 256]). In [8] Mutylin gave an example of a locally bounded nonnormable field extension of the p-adic number field.…”

Norms onF(X)

“…We note that in each case the author has actually assumed that F is bounded.) In this paper we modify a technique of Mutylin [6] to show that such a characterization is impossible by constructing for each infinite field F, a norm || 11 on F(X) for which F is discrete, \\X\\ < 1, the completion of F(X) is a field but || || is not equivalent to the supremum of any finite family of absolute values on F(X). In the process, we also obtain a norm || || on the polynomial ring F [X] Proof.…”