The following is a powerful powerful theorem: Let be a simple field extension of field , where is algebraic over . Then every is also algebraic over . Also, .
The proof I think is most brilliant. I am not going to provide you with a proof here. I am just going to try and slightly generalize this theorem.
Let . Consider the vector space generated by . Also, let be the set of vectors that span the vector space generated by . Then .
How is this a generalization? Firstly, the set of vectors need tot span the whole of . It only needs to span the subspace generated by . Hence, we can find a better upper bound for the degree of the irreducible polynomial satisfying . Secondly, we need only consider the set of vectors that spans the space, and not necessarily the basis of the space. Although this is likely to give us a worse estimate for the upper bound of the irreducible polynomial, it is still a generalization (sometimes, it may be impractical to determine the basis from the set spanning the vector space).