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Classically, the ring of integers of a number field $K / \mathbb{Q}$ is defined to be the collection (forms a ring) of those elements $\alpha \in K$ such that there is a monic $f \in \mathbb{Z}[x]$ satisfied by $\alpha$.

I am just beginning to study local fields, and I saw that the ring of integers of a local field $K$ is defined to be the closed unit ball of $K$. I'm concerned about why the same terminology is used. It is not obvious to me why this is a natural extension of the ring of integers of a number field.

Would someone be able to explain the motivation for using the same terminology?

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  • $\begingroup$ Please have a look at my last two paragraphs ;-) $\endgroup$ – Watson Jun 28 '18 at 15:23
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The ring of integers of a non-archimedean local field $K$ of characteristic $0$ (i.e. an extension of $\Bbb Q_p$ for a unique prime number $p$) is the integral closure of $\Bbb Z_p$ in $K$. This is the same situation as for number fields: when $L$ is a number field, the ring of integers $O_L$ is the integral closure of $\Bbb Z$ in $L$.

We can further notice that $\Bbb Z$ is dense in $\Bbb Z_p$ (and even in $\hat{\Bbb Z}$). The $p$-adic unit ball $\Bbb Q \cap \Bbb Z_p$ in $\Bbb Q$ is the localization $\Bbb Z_{(p)}$, which is a DVR (hence PID, Dedekind, UFD, integrally closed, ...). But the ring of integers $\Bbb Z$ is not a "unit ball" ; it is rather an intersection of the unit balls $\Bbb Z_{(p)}$ with $p$ varying among the primes: $$\Bbb Z = \bigcap_p \Bbb Z_{(p)}.$$ This can be generalized to the ring of integers of any number field, as shown here.


In general, if $(K,v)$ is a valued field, we can define the valuation ring as $O_K = \{x \in K \mid v(x) \geq 0\}$. In particular, $$\Bbb Z_p = \{ x \in \Bbb Q_p \mid v_p(x) \geq 0 \},$$ and notice that $v_p(x) \geq 0 \iff |x|_p \leq 1$. The condition $v_p(x) \geq 0$ means that $x$ has a "denominator" not divisible by $p$ (this makes sense at least if $x \in \Bbb Z$). Any other prime $q \neq p$ is invertible in $\Bbb Z_p$, so intuitively the condition $v_p(x) \geq 0$ should mean that $x$ has no denominator, i.e. should be considered as a ($p$-adic) integer.

It might be useful to mention that $\Bbb Q_p = \Bbb Z_p[1/p]$ (as algebras), and any $z \in \Bbb Q_p^{\times}$ can be written uniquely as $z = p^n u$ with $u \in \Bbb Z_p$ and $n \in \Bbb Z$. The condition $|x|_p \leq 1$ (equivalently $v_p(x) \geq 0$) just means that $n \geq 0$, whereas $n = -1$ for instance gives $z = u / p$, which is not a $p$-adic integer, having $p$ as "denominator".

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