Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

That is, if a function $f$ is analytic and bounded in all $K$, a $p$-adic field (or more generally a complete non-archimedean field), has to be constant? And does the theorem work for functions on $K^n$, or in $\mathbb{C}^n$?

share|cite|improve this question
"Analytic" means what? Since the space is zero-dimensional, there are certainly sets $U$ that are open and closed, not empty and not the whole space. The characteristic function of such a set is identically constant on a neighborhood of every point. Note that the answer says "entire" and not "analytic"... – GEdgar Jan 14 '12 at 14:46
up vote 3 down vote accepted

The answer to the first question is yes, Liouville's theorem still holds for valued fields that are algebraically closed (this last part added after Pete Clark's comment below). See, for example, these lecture notes by William Cherry (in particular, see page 16).

share|cite|improve this answer
Note that Cherry's result is for a valued field which is algebraically closed, e.g. $\mathbb{C}_p$. If I remember correctly, there are nonconstant bounded entire functions on $\mathbb{Q}_p$ (or any finite extension thereof), just as there are on $\mathbb{R}$. I believe Alain Robert's GTM has a discussion of this. – Pete L. Clark Jan 14 '12 at 5:01
In fact, in any normed field, the nonexistence of nonconstant bounded entire functions implies that every nonconstant polynomial has a root, so algebraic closure is necessary as well as sufficient. – Pete L. Clark Jan 14 '12 at 8:34
Thanks, I just need the case of an algebraically closed field. – iago Jan 14 '12 at 12:26

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.