TITLE: Reference request: representations of unipotent groups have a fixed point.
QUESTION [6 upvotes]: I'm looking for a reference for the following standard result: 
Let $U$ be a unipotent algebraic group over an algebraically closed field $k$ (of any characteristic); then any algebraic representation of $U$ has a fixed point. 
Statements of Engel's theorem for the analogous statement about Lie algebras seem to be ubiquitous. I can also find the statement that connected solvable groups always preserve a line in many places (for example, Borel, Theorem III.10.4). Combining this with the fact that unipotent groups have no non-trivial characters gives me the result I need. But it would be nice to have a place to which I could refer for the precise statement about unipotent groups.

REPLY [5 votes]: Theorem 17.5 in Humphreys's Linear Algebraic Groups seems to be the result you want. (Also, doesn't the result for solvable groups only imply the corresponding result for connected unipotent groups? The proof for unipotent groups doesn't require connectedness.)

REPLY [5 votes]: This is Kolchin's theorem, first proved in Kolchin's paper http://www.jstor.org/stable/1969399