Suppose one doesn't know any Galois theory, but they are familiar with the basic theory of field extensions. Then how would one justify the following picture?
I'm pretty sure they're just saying these are the known subfields, and there could be more but they just write down those ones (which seems weird, which is why I'm asking). Or have they deduced that this is a comprehensive list of subfields just by using basic theory? This occurs on page 573 of Dummit and Foote's Abstract Algebra.