Quotient fields of a model of \(I\Delta_0+\Omega_1\) (Q2743645)

\(I\Delta_0\) stands for ``\(\Delta_0\) (= bounded) induction'', of course, and \(\Omega_1\) is the sentence ``\(x^{[\log y]}\) exists for all \(x,y>0\)''. The main result of this paper reads: Any quotient field of a model of \(I\Delta_0+ \Omega_1\) has a unique extension of each finite degree. In the joint paper with \textit{A. MacIntyre} [Isr. J. Math. 117, 311-333 (2000; Zbl 0993.03076)], the author shows the same result for the case when the quotient field is given by the principal ideal generated by a prime. Here, the result is extended to any maximal ideal. On the way to showing the above, a number of interesting results are established. For instance, Peano arithmetic is shown to be equivalent to \(I\Delta_0+\)``every definable prime ideal is principal''. This paper is self-contained and readable, for ample background material is provided, leitmotifs are explained, and so forth.