E⊂F⊂K are nonalgebraic field extensions. α∈K∖F is algebraic over F. L is generated over E by coefficients of the monic minimal polynomial of α over F. Which statements hold?
Correct answer: (A) F(α)/L(α) is finite iff F/L is finite.; (D) F(α)/L(α) is algebraic iff F/L is algebraic.
Explanation
The minimal polynomial has coefficients in L and remains irreducible there: a factorization over L would factor it over F. Thus [L(α):L]=[F(α):F]=m, refuting B,C. If F/L is finite, adjoining α preserves finiteness; conversely F(α)/L(α) finite and L(α)/L finite imply F/L finite by the tower law. The analogous argument using transitivity/subextensions of algebraic extensions proves D. **A,D**.
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