On the Representation of Fields as Finite Sums of Proper Subfields.

We study which fields F can be represented as finite sums of proper subfields. We prove that for any n ≥ 2 every field F of infinite transcendence degree over its prime subfield can be represented as an unshortenable sum of n subfields, and every rational function field F = K (x 1 , … , x n) can be represented as an unshortenable sum of n + 1 subfields. We also show that no subfield of the algebraic closure of a finite field is a finite sum of proper subfields, and no finite extension of the field Q of rationals can be decomposed into a sum of two proper subfields. [ABSTRACT FROM AUTHOR]