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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.

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

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