THEOREM: Let be a field of elements contained in an algebraic closure of . The elements of are precisely the zeros in of the polynomial in .
PROOF: The set of non-zero elements of forms a multiplicative group of order under the field multiplication. For , the order of in this group divides . Thus for we have so . Therefore every element of is a zero of . Since can have atmost zeros, we see that contains precisely the zeros of in .
QUERY: Where have we used the algebraic closureness of . Wouldn't the proof still work if instead of we had any field which is an extension field of the field containing the field .
May be i am missing the obvious but can't figure it out.