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

In Atiyah & MacDonald they provide an abstract way to construct valuation rings, I'm curious how easy it is to work this out in general. As a refresher let $K$ be a field, $L$ an algebraically closed field and $\mathcal V$ be the set of ordered pairs $(A,f)$ where $A$ is a subring of $K$ and $f: A \rightarrow L$ is a homomorphism. In particular ordering $\mathcal V$ by $(A,f) \leq (A^\prime,f^\prime)$ if $A \subset A^\prime$ and $f^\prime |_A=f$, Atiyah & Macdonald show any maximal element of $\mathcal V$ is a valuation ring.

I was able to show that when $K=\mathbb Q$ and $L=\bar{\mathbb Z_p}$ that $\mathbb Z_{(p)}$ is a maximal element, are there any others? What are the maximal elements if we just take $K=\mathbb Q$ and $L=\bar{\mathbb Q}$. What if $K=\mathbb R$ and $L=\bar{\mathbb Z_2}$?

share|cite|improve this question
What is $\bar{\mathbb Z_p}$ ? Do you mean $\overline{\mathbb Q_p}$ ? – user18119 Jan 17 '13 at 22:30
@QiL No, I mean the algebraic closure of the field with $p$ elements. – JSchlather Jan 18 '13 at 1:17
Dear Jacob, I think this is a very bad notation. – user18119 Jan 18 '13 at 8:04
@QiL How do you denote the algebraic closure of the field with $p$ elements? – JSchlather Jan 18 '13 at 15:13
I meant $\mathbb Z_p$ usually denotes the ring of $p$-adic integers. For the field in $p$ element, $\mathbb F_p$ is the standard notation. – user18119 Jan 19 '13 at 20:16
up vote 2 down vote accepted

Note that the construction in A-M gives rise to a valuation ring whose maximal ideal is the kernel of $f: A\to L$. In particular, if $L$ has characteristic $p$, then the valuation of $p$ is positive or infinite.

If $K=\mathbb Q$, by Ostrowski's theorem, any valuation on $K$ is $\ell$-adic for some prime $\ell$. The above remark implies that the construction of A-M gives the p-adic valuation on $\mathbb Q$.

If $K=\mathbb Q$ and $L=\overline{\mathbb Q}$ or any field of characteristic $0$, there is a unique ring homomorphism from $K$ to $L$ and it is the injective. So the maximal element is just $A=\mathbb Q$ corresponding to the trivial valuaton.

When $K$ is any field of characteristic $0$ and $L$ has characteristic $p>0$, then the maximal elements are extensions of the $p$-adic valuation on $\mathbb Q$. As soon as $K$ has a transcendental element over $\mathbb Q$, there are infinitely many such valuations. Indeed, this is true over a purely transcendental subfield $\mathbb Q(t)$ of $K$ (valuation rings : $\mathbb Q[t]_{(t-r)\mathbb Q[t]}$, for any $r\in \mathbb Q$), and A-M's construction shows that any valuation of $\mathbb Q(t)\subseteq K$ extends to $K$.

share|cite|improve this answer

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.