What is an example of a finite field extension which is not generated by a single element?


Background: A finite field extension E of F is generated by a primitive element if and only if there are a finite number of intermediate extensions. See, for example, [Lang's Algebra, chapter V, Theorem 4.6].



Finite extension of fields with no primitive element
Let $F$ be a finite field with $p$ elements. Let $K=F(x,y)$ be the field of rational functions in two indeterminate variables over $F$. Consider the extension of $K$ obtained by adjoining $p$-th roots of $x$ and of $y$. More precisely, let $k$ be an algebraic closure of $K$. In $k$ we can solve the equation $X^p=x$ in the variable $X$. Let $a$ be a solution of this equation; so $a$ is an element of $k$ which satisfies $a^p=x$. Similarly find an element $b$ which satisfies $b^p=y$. 


Consider $L=K(a,b)$. $L$ is a finite extension of $K$, of order $p^2 $ as you can check. However there is no element of degree $p^2$ in $L$, and a primitive element would have to have degree $p^2$. 


This example is, in a sense, the simplest possible. Separable finite extensions are simple (contain a primitive element), so we must use a non-perfect base field. Also, extensions of degree $p$ are also simple, so we must use $p^2$. 


