On the representation of fields as sums of two proper subfields

In this paper, we study which field [Formula: see text] can be represented as sums of two proper subfields. We prove that a field [Formula: see text] is a sum of two proper subfields if and only if the transcendence degree of [Formula: see text] over its prime subfield is infinite. This answers a question posed in the paper [M. Kȩpczyk and R. Mazurek, On the representation of fields as finite sums of proper subfields, Results Math. 75(2) (2020) 65].