Let [itex] K \subseteq F [/itex] be fields and let R be a ring such that [itex] K \subseteq R \subseteq F [/itex]. If F is algebraic over K, show that R is a field. If F is algebraic over K, then every element of F is a root of some polynomial over K[x]. But since K is contained in R, every element of F is thus a root of some polynomial over R[x]. I want to show that every nonzero element of R has an inverse which would show that it is a field. The elements in K are obviously invertible so I need to show that any element in R that is not in K is also invertible. I am having trouble with this part and can't think of a way to show this. Can someone offer a hint or two in the right direction? Help is greatly appreciated.