# Degree $n$ extension of local field splits degree $n$ division algebra

I am trying to write an article which is pretty self-contained on the number theory side, and would like to use the following result:

Let $K$ be a local field, $n > 1$ a natural number, $D$ a division algebra of degree $n$ over $K$. For any degree $n$ extension $L/K$, $D \otimes L$ is split.

As far as I am aware, this result is classically derived as a corollary from the description of the Brauer group of $K$ as $\mathbb{Q}/\mathbb{Z}$ (if $K$ is non-archimedean; the case $K = \mathbb{R}$ is easy). This is how it is done for example in Pierce's Associative Algebras or Weil's Basic Number Theory, as well as some documents I found online.

This is the only result on central simple algebras over local fields I need; I do not need the full strength of the isomorphism to $\mathbb{Q}/\mathbb{Z}$. Is there a short argument, using only basic properties of local fields? If that makes things easier, I may even assume $n$ to be prime.

• I think it is hard to make it more "elementary" than Weil's BNT argument, and yet avoid incomprehensible tricks. – paul garrett Mar 8 '18 at 21:18
• This answer to my closely related question refers to Serre's Local fields, chapter 13. Not sure that really gives what you seem to be looking for :-/ – Jyrki Lahtonen Mar 9 '18 at 5:31
• One option is to just give a proof when $n=2$, and say $F=\mathbb Q_p$, $p$ odd, where you can do things by hand. – Kimball Mar 9 '18 at 12:48
• @Kimball Agreed that the case $n = 2$ can be done by elementary quaternion algebra arguments, but these do not generalise easily to other primes, do they? – Bib-lost Mar 16 '18 at 14:22
• No, I don't think so. I expect that you can give a similar argument if you take for granted that division algebras over local fields are cyclic, but if you want to give a complete argument, I don't know an easy way. – Kimball Mar 16 '18 at 17:17