proof of primitive element theorem
Let and be fields, and let be finite.
Suppose first that . Since is finite, is algebraic over . Let be the minimal polynomial of over . Now, let be an intermediary field with and let be the minimal polynomial of over . Also, let be the field generated by the coefficients of the polynomial . Thus, the minimal polynomial of over is still and . By the properties of the minimal polynomial, and since , we have a divisibility , and so:
Since we know that , this implies that . Thus, this shows that each intermediary subfield corresponds with the field of definition of a (monic) factor of . Since the polynomial has only finitely many monic factors, we conclude that there can be only finitely many subfields of containing .
Now suppose conversely that there are only finitely many such intermediary fields . If is a finite field, then so is , and we have an explicit description of all such possibilities; all such extensions are generated by a single element. So assume (and therefore ) are infinite. Let be a basis for over . Then . So if we can show that any field extension generated by two elements is also generated by one element, we will be done: simply apply the result to the last two elements and repeatedly until only one is left.
So assume . Consider the set of elements for . By assumption, this set is infinite, but there are only finitely many fields intermediate between and ; so two values must generate the same extension of , say and . This field contains
and so letting , we see that