Suppose we have an algebraic group $G$ defined over a field $k$. Suppose we consider the fraction field of $k[G]$. Is it possible to get a situation where this field is not a separable extension of $k$? An example would be very helpful.



an algebraic group where the function field is not separable over the ground field
The answer depends on what you mean by "algebraic group". If you mean algebraic group in the sense of Borel/Humphreys/Springer, the field of fractions of $k[G]$ is always separably generated, more-or-less by definition. If you mean algebraic group scheme, which must be integral for the question to make sense, then it doesn't have to be separably generated, as shown by Tom's example.


