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Suppose F $\subset$ L is a field extension, a, b $\in$ L are algebraic over F. Prove that [F(a, b): F] is finite.

Unfortunately I don't even know where to begin with this one, other than establishing the tower of extensions:

F $\subset$ F(a) $\subset$ F(a, b)

What does $F(a)$ even look like? Is it $F(a) =$ $\{ u + v\cdot a$ | $u, v \in F\}$ ?

Thanks so much for the help!

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Hints:

  1. What does it mean that $a$ is algebraic over $F$?
    (Note that $F(a)=\{u+v\cdot a\,\mid\,u,v\in F\}$ only if the degree of $a$ over $F$ is $2$ [or if $a\in F$ already].)
  2. Is $b$ also algebraic over $F(a)$?
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