# If L is module-finite over K, and K ⊂ R ⊂ L, then R is a field [duplicate]

I am self studying, and this question is from Fulton's Algebraic Curves (question 1.50 (b))

1.50∗. Let K be a subfield of a field L.
(a) Show that the set of elements of L that are algebraic over K is a subfield of L containing K. (Hint: If $$v^n +a_1v^{n−1} +···+a_n = 0$$, and $$a_n \neq 0$$, then $$v(v_{n−1} +···) = −a_n$$.)
(b) Suppose L is module-finite over K, and K ⊂ R ⊂ L. Show that R is a field.

I could do (a) but I cannot do (b). I cannot think of any specific examples, and don't know how to begin the proof. It seems like there needs to be at least some sort of condition on R (like that it is a subring of L), but that is not given.

• $R$ is definitely a ring. Can you solve it if you assume that $R$ is a ring? – jgon Sep 5 '19 at 6:17
• @jgon Oh okay, I got it. Thank you. – rr01 Sep 5 '19 at 6:49

Without assuming $$R$$ is a subring of $$L$$, there's no way to prove it's a field. Example: just add to $$K$$ an element in $$L$$, but not in $$K$$; this set is definitely not a subfield of $$L$$.
Assuming $$R$$ is a subring of $$L$$, the only thing to prove is that every nonzero element of $$R$$ has an inverse in $$R$$. Suppose $$a\in R$$, $$a\ne0$$.
If $$n=\dim_K L$$ (a finite module over $$K$$ is a finite dimensional $$K$$-vector space), then $$1,a,\dots,a^n$$ are not linearly independent. Can you finish?