TITLE: Is there a field which is the union of finitely many proper subfields?
QUESTION [6 upvotes]: Is there a field which is the union of finitely many proper subfields?

REPLY [4 votes]: See A Bialynicki-Birula, J Browkin, A Schinzel, On the representation of fields as finite unions of subfields, Colloq Math 7 (1959) 31-32, MR 22 #2601. At the risk of violating copyright, here's the review, by S W Golomb:
This article contains a short proof of the fact that no field can be represented as a finite union of proper subfields. A counterexample is exhibited to the corresponding assertion for integral domains.
EDIT: The paper may be available at http://matwbn.icm.edu.pl/ksiazki/cm/cm7/cm717.pdf