Let be an algebraic field extension. Recall that an element is called **separable** over if the minimal polynomial of over has no repeated root (in its splitting field). Otherwise, is called** inseparable**. If every element of is separable over then is called a **separable extension** of If every element of is inseparable over then is called a **purely inseparable extension** of Recall also the following facts from basic field theory.

**Fact 1**. If then is separable. If then an element is separable over if and only if the minimal polynomial of over is not in

**Fact 2**. If and then is purely inseparable if and only if for every there exists some integer such that

**Theorem**. (Jacobson, Noether) Let be a noncommutative division algebra with the center If is algebraic over then there exists an element of which is separable over

*Proof*. If let and put Then, by Fact 1, is separable and we are done. Suppose now that and every element of is inseparable over Then for every the field is purely inseparable over So, by Fact 2, for some integer Now, fix an element and an integer such that Define the map by for all Clearly where are the -linear maps defined by and for all Since and we have

Therefore for all because Hence

Now, since we have and so for some Thus, by there exists an integer which is maximal with respect to the property Hence and so if we let then and i.e. is invertible and it commutes with Let See that and so By Fact 2, there is an integer such that Then

which is non-sense.