Uncountable Fields Have Proper Uncountable Subfields

Abstract

In a first encounter of the study of fields in an abstract algebra course, a student learns about various subfields of R1 and C which are usually obtained by adjoining a finite number of elements to the field G. The student may also learn about the subfield of lR consisting of those real numbers which are algebraic over G. Each of these subfields of lR has the property that it contains a countable number of elements. Since the field lR is uncountable, the question naturally arises as to whether there exists an uncountable proper subfield of lR. An affirmative answer to this may be given by appealing to the existence of a transcendental basis of lR over a [1]. The idea of a transcendental basis is not commonly encountered in an undergraduate abstract algebra course, but many students do encounter Zorn's lemma with which one can construct an elementary, yet interesting, proof of the following proposition.