Thanks to everyone for the help! However, I am not familiar with any theorems which show that algebraic extensions "behave well in towers." After a little bit more thinking, I have a solution which I think fits better with the methods of my text. I reproduce it here just in case anyone feels like checking it for errors:

*Proof:* Let

with

, and assume towards a contradiction that

is algebraic over

. Then there are

with

, and

. Now,

has the form

,

where

, and each

. Consider the polynomial

. [EDIT: Assuming this polynomial is nonzero,] we have

,

such that

is algebraic over

. However, this contradicts the hypothesis that

is transcendental over

. So our assumption must be false, and the conclusion follows.