## The Primitive Element Theorem

### The Primitive Element Theorem

If F is finite over K, F is a simple extension K(u) iff F has a
finite number of intermediate fields.
Here u is the "primitive element".
This theorem is true if K is a finite field,
so assume K is infinite.
Let F/K have a finite number of intermediate fields.
Choose u in F so that the dimension of K(u) over K is maximal.
Suppose this remains a proper subfield of F,
and draw v from F-K(u).
Consider all intermediate fields K(u+av) where a is taken from K.
Since K is infinite, there must be a distinct pair a and b
such that K(u+av) = K(u+bv).
It follows that (u+av)-(u+bv) = (a-b)v is in K(u+av).
Divide by a-b, multiply by a, and subtract from u+av to get u.
And since we have u, we also have v.
Thus K(u+av) properly contains K(u),
and pulls in v,
which contradicts the selection of u.
There must be some u with K(u) = F, and F is a simple extension.

Conversely, assume F = K(u).
Let u satisfy the monic irreducible polynomial p(x).
If E is an intermediate extension, factor p(x) in E,
and find the irreducible factor q(x) such that q(u) = 0.
Adjoin the coefficients of q to K to obtain a subfield of E.
Call this subfield L, and suppose L is properly contained in E.
Of course q remains irreducible over L, and q(u) is still 0.
The dimension of L(u) over L is the degree of q.
Yet, the dimension of E(u) over E is also the degree of q,
and L(u) and E(u) are both F.
Therefore E and L have the same dimension over K,
and since one contains the other, they are equal.
Our subfield E is generated by the coefficients of the irreducible polynomial q(x).

The polynomial p(x) factors uniquely in the ring F[x].
These prime factors clump together in finitely many ways to build intermediate polynomials q(x).
Therefore there are finitely many intermediate fields.

As a corollary, separable finite extensions are simple.
Let F/K be such an extension and let E/K be its normal closure.
Now E is galois and finite, with galois group G.
Intermediate extensions correspond to the subgroups of G,
and there are finitely many of these.
Therefore there are finitely many extensions between F and K,
and F/K is a simple extension K(u) for some primitive element u.