We have proved that every element `beta in F(alpha)` such that `beta ne F` must be transcendent over F.
The proof utilizes a strategy of contradiction. By assuming an element `beta in F(alpha)` where `beta ne F` is algebraic over F, a chain of logical steps leads to a contradiction.
What is the main contradiction that drives this proof and ultimately confirms the desired statement:
A) The initial assumption that `beta ne F` is contradicted by showing that `beta` must necessarily be an element of F.
B) The initial assumption that `beta` is algebraic over F is contradicted by showing that `alpha` must be algebraic over F ?