On irregular semi strong P-ADIC U numbers

Abstract

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The concept of the "relation of comparability" was introduced by Maillet in [7], who showed that if $\alpha$,$\beta$ are comparable Liouville numbers then each of the numbers $\alpha+\beta$, $\alpha-\beta$, $\alpha\beta$ and $\alpha/\beta$ is either a rational or Lioville number. Moreover those which are Liouville numbers are comparable among theem and too and p. Maillet's proof uses in an essential way the transitivity of the comparability relation. Unfortunately, as the comparability relation is not transitive, his proof is defective. In this paper, without using the comparability relation, we obtain some uncountable subfields of p-adic numbers field, Qp. In [1] using a different notion of comparability, Alnıaçık was able to define some uncountable subfields of C. In this paper, without using comparability relation, we define irregular semi-strong p-adic $U_m$ numbers and obtain some uncountable subfields of p-adic numbers field Qp