Suppose G is a connected affine algebraic group over a field k. If G is reductive or k is perfect, then G is unirational. A reference is Borel, Linear Algebraic Groups, Theorem 18.2. This doesn't quite give rationality, but is the best result in general that I know of, and is sufficient for some applications. (Ryan's answer of course gives more precise information under stronger hypotheses).

An affine counterexample over an imperfect field of characteristic p is given by y^p=x+tx^p where t is not a p-th power in k. (This is a subgroup of G_a²).

Now if G is connected and not affine, then it surjects onto an abelian variety. So if G were rational, then we could get a nonconstant morphism from P^1 to an abelian variety. Now let me just quote http://math.stackexchange.com/a/114611 so no non-affine conntected G can be rational, or even unirational.

Suppose G is a connected affine algebraic group over a field k. If G is reductive or k is perfect, then G is unirational. A reference is Borel, Linear Algebraic Groups, Theorem 18.2. This doesn't quite give rationality, but is the best result in general that I know of, and is sufficient for some applications. (Ryan's answer of course gives more precise information under stronger hypotheses).

An affine counterexample over an imperfect field of characteristic p is given by y^p=x+tx^p where t is not a p-th power in k. (This is a subgroup of G_a²).

Now if G is connected and not affine, then it surjects onto an abelian variety. So if G were rational, then we could get a nonconstant morphism from P^1 to an abelian variety. Now let me just quote https://math.stackexchange.com/a/114611 so no non-affine conntected G can be rational, or even unirational.

Suppose G is a connected affine algebraic group over a field k. If G is reductive or k is perfect, then G is unirational, which implies rationality. A reference is Borel, Linear Algebraic Groups, Theorem 18.2.

An affine counterexample over an imperfect field is given by y^p=x+tx^p where t is not a p-th power in k. (This is a subgroup of G_a²).

Now if G is connected and not affine, then it surjects onto an abelian variety. So if G were rational, then we could get a nonconstant morphism from P^1 to G. Now let me just quote http://math.stackexchange.com/a/114611 so no non-affine conntected G can be rational.

Suppose G is a connected affine algebraic group over a field k. If G is reductive or k is perfect, then G is unirational. A reference is Borel, Linear Algebraic Groups, Theorem 18.2. This doesn't quite give rationality, but is the best result in general that I know of, and is sufficient for some applications. (Ryan's answer of course gives more precise information under stronger hypotheses).

An affine counterexample over an imperfect field of characteristic p is given by y^p=x+tx^p where t is not a p-th power in k. (This is a subgroup of G_a²).

Now if G is connected and not affine, then it surjects onto an abelian variety. So if G were rational, then we could get a nonconstant morphism from P^1 to an abelian variety. Now let me just quote http://math.stackexchange.com/a/114611 so no non-affine conntected G can be rational, or even unirational.