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).