There is a well-known structure theorem for locally compact non discrete topological division algebras, see here


<https://math.stackexchange.com/q/1160086/187521>


(I repost it here because I think it is more suitable given the nature of the question) and the proof of this theorem generally always uses the existence of Haar measures on locally compact topological groups. Intuitively I don't see how we could go without this existence, but I would like to be sure. *Could we prove the structure theorem without Haar measure or not ?*



Structure of locally compact non discrete topological division algebras without the use of Haar measure
I am a little surprised that you think it's "intuitive" that Haar measure *must* be used.


In section 9.13 of Jacobson's "Basic Algebra II" this result is proved for totally disconnected division algebras. In section 27 in Chapter IV of Warner's "Topological Fields" Theorem 27.2 covers the connected case. Neither book uses Haar measure.


