Let $Q$ be a quiver without loops (cycles of length 1). Kac proved that if $K$ is algebraically closed, the dimension vectors of indecomposable representations of $Q$ over $K$ are exactly the positive real roots and the imaginary roots; further, there is a single isomorphism class of representations if the root is real, and infinitely many if the root is imaginary.
My question is: what is known if we drop the assumption that $K$ is algebraically closed? For real roots, this was answered by Schofield (The field of definition of a real representation of a quiver $Q$. Proc. AMS 116 (1992), no. 2, 293--295) --- he showed one can drop the assumption and nothing changes.
My go-to textbook for quiver representations over non-algebraically closed ground fields (Deng, Du, Parshall, Wang, "Finite dimensional algebras and quantum groups") in this case only gives the result for algebraically closed fields.
Edited to add: I am writing an expository note which mostly works over an arbitrary ground field, but I would also like to mention Kac's theorem, so I feel like it's incumbent on me to say something about what the state of our knowledge is about Kac's theorem for arbitrary ground fields. I guess mainly my focus is on the question of what vectors can appear as dimension vectors of indecomposable representations --- is even that much known generally?
Also: there is a remark by Jan Schröer here from 2016 to the effect that in full generality, we do not have an analogue of Kac's theorem.
