>>13368754Don't have a zero IQ. First thing to do is to realize that these elements, combined with form a spanning set for .
Next, note that is the splitting field of . Since this polynomial is irreducible via Eisenstein's criterion, it follows that
Thus, as the initial three elements were part of a four element spanning set, it follows that our four elements were a basis and must therefore be linearly independent. I'm not going to reprove basic field theory theorems right now for you however.
