Transitivity of Algebraic Extensions

Let K\subseteq E\subseteq F be a tower of fields. If F/E and E/K are algebraic, then F/K is algebraic.

Proof:

(Hungerford pg 237, reworded)

Let u\in F. Since u is algebraic over E, there exists some b_i\in E (b_n\neq 0) such that \displaystyle b_0+b_1u+\dots+b_nu^n=0.

Let L=K(b_0,\dots,b_n), then u is algebraic over L. Hence L(u)/L is finite. Note that L/K is finitely generated and algebraic, since each b_i\in E is algebraic over K. Thus L/K is finite.

Thus by Tower Law, L(u)/K is finite, thus algebraic.

Hence u\in L(u) is algebraic over K. Since u was arbitrary, F is algebraic over K.

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