Primitive element theorem

Statement

Let $L/K$ be a finite separable extension. Then $L/K$ is a simple extension: there is $\theta \in L$ with $L = K(\theta)$.

Remarks

Every finite extension of a field of characteristic $0$ is separable, so it is simple. For example, $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2} + \sqrt{3})$. For an infinite field $K$, the proof shows that $\theta = a + c b$ works for all but finitely many $c \in K$. For finite fields, a primitive element of $L$ generates $L$ over $K$. Without separability the statement can fail: there are finite extensions of degree $p^2$ in characteristic $p$ with no primitive element.

History

Évariste Galois sketched a proof for splitting fields over the rationals in his First Memoir of 1831, published in 1846. Ernst Steinitz proved the theorem in its modern form in 1910. Emil Artin later developed Galois theory without primitive elements.

Explanations

L1 human

A finite separable extension, even one built from many elements, can be generated by one element. So many statements about finite extensions reduce to the case of one algebraic element and its minimal polynomial.