Proving Field Properties of R if F is Algebraic over K

[demonelite123]

Let $K \subseteq F$ be fields and let R be a ring such that $K \subseteq R \subseteq F$. 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.

 

[morphism]

Let a be a nonzero element of R. Here are two independent (but ultimately equivalent) hints to help you show that the inverse of a belongs to R.

Hint A: K(a)=K[a].

Hint B: Write down the inverse of a in F, using the fact that a is algebraic/K.