# On the structure of non discrete locally compact topological (non-necessarily commutative) complete fields.

There is totally classic result about the structure of non discrete locally compact topological (non-necessarily commutative) fields $K$, whose proof uses the existence of the Haar measure on the underlying additive group of $K$. The theorem (see Bourbaki, Commutative algebra, chapter VI, paragraph 9, section 4, thorem 1 for instance) is the following :

Theorem. Let $K$ be a locally compact non discrete (not necessarily commutative) field. Note $|\cdot|$ the modulus of $K$ (defined thx to the Haar measure), which is an absolute value on $K$.

1. If $K$ is of characteristic $0$ and if $|\cdot|$ is not non-archimedean, then $K$ is isomorphic ot $\mathbf{R}$, $\mathbf{C}$ or $\mathbf{H}$.

2. If $K$ is of characteristic $0$ and if $|\cdot|$ is non-archimedean, then $K$ is an algebra of finite rank of the field $\mathbf{Q}_p$ for some prime number $p$.

3. If $K$ is of charateristic $p > 0$ ($p$ is prime) it is isomorphic to a field whose center is a field $k((T))$ where $k$ is finite extension of $\mathbf{F}_p$, and $K$ is of finite rank over its center.

The proof of the existence of a (left or right) Haar measure on a locally compact topological group is not that hard, but I would nevertheless like to know if it is possible to derive the structure of non discrete locally compact topological (non-necessarily commutative) fields without using the Haar measure.

Remark 1. All compact spaces are supposed to be Hausdorff.

Remark 2. Bourbaki defines a discrete field as we all can imagine (in Topologie générale, chapter 3, p. 55 examples, 2) as a topological field whose topology is the discrete one, and I guess that in the aforementioned theorem the "non discrete" means "whose topology is not the discrete one". Then in the case 2 with simply $K = \mathbf{Q}_p$ of case $3$ with simply $K = \mathbf{F}_p ((T))$, $K$ may indeed by a totally disconnected topological space, but it is not a discrete topological space.

• The connected ones are the reals, the complex numbers and the quaternions. What are all non-discrete ones? Just curiosity. – Henno Brandsma Feb 22 '15 at 13:39
• All finite extensions of $\mathbf{Q}_p$'s or of $\mathbf{F}_p ((T))$ for instance. And only those. – Olórin Feb 22 '15 at 13:42
• Are the reals a finite extension of $\mathbf{Q}_p$ or $\mathbf{F}_p((T))$? For which $p$? – Henno Brandsma Feb 22 '15 at 13:44
• Ok, it will be easier to detail the structure theorem in my question, which I will do now – Olórin Feb 22 '15 at 13:58
• So to answer you first question properly : the only connected ones are those you mentioned, and the others (non-connected and in fact totally disconnected) commutative ones are finite extensions of $\mathbf{Q}_p$'s or of $\mathbf{F}_p$'s, for some prime $p$ which is the caracteristic of the residue field (and also of the field it it is of char. $> 0$.) – Olórin Feb 22 '15 at 14:21