primitive element theorem

If you adjoin two algebraic numbers to a field, you might expect to need both of them forever to describe the result. The primitive element theorem is a pleasant surprise: under a mild hypothesis, a single cleverly chosen element already generates everything. Instead of carrying around K(a, b), you can find one number c with K(a, b) = K(c). A complicated extension collapses to a simple one.

Precisely, every finite separable extension L of K is simple: there exists a primitive element c in L with L = K(c). The usual construction takes c = a + lambda b for a suitable scalar lambda in K, avoiding the finitely many bad values; this works whenever K is infinite, and finite fields are handled separately since their multiplicative groups are cyclic.

The separability hypothesis is essential. There exist finite extensions that are not simple — the classic counterexample is F_p(s, t) over F_p(s^p, t^p), an inseparable extension of degree p^2 that needs two generators. So the theorem is really a statement about separable extensions, and it underlies the convenience of working with a single minimal polynomial in Galois theory.