- What is a valuation on a field \(K\) (specifically a discrete valuation)?
- Define the \(p\)-adic valuation \(v_p\) and the \(p\)-adic absolute value \(|\cdot|_p\) on \(\mathbb{Q}\).
- Define the ring of \(p\)-adic integers \(\mathbb{Z}_p\) and the field of \(p\)-adic numbers \(\mathbb{Q}_p\) (both algebraically via inverse limits and analytically via completion).
Solution
A discrete valuation on a field \(K\) is a map \[ v:K\to\mathbb Z\cup\{\infty\} \] such that \[ v(x)=\infty\iff x=0,\qquad v(xy)=v(x)+v(y),\qquad v(x+y)\ge\min\{v(x),v(y)\}, \] and whose image on \(K^\times\) is all of \(\mathbb Z\). Its valuation ring is \[ \mathcal O_v=\{x:v(x)\ge0\}, \] with maximal ideal \[ \mathfrak m_v=\{x:v(x)>0\}. \]
For a prime \(p\), every nonzero \(x\in\mathbb Q\) has a unique form \[ x=p^k\frac ab, \qquad p\nmid a,b, \] and one defines \[ v_p(x)=k,\qquad v_p(0)=\infty. \] The associated absolute value is \[ |x|_p=p^{-v_p(x)},\qquad |0|_p=0, \] and satisfies \[ |x+y|_p\le\max\{|x|_p,|y|_p\}. \]
The field of \(p\)-adic numbers is the metric completion \[ \mathbb Q_p=\widehat{\mathbb Q}^{\,|\cdot|_p}. \] Its valuation ring is \[ \mathbb Z_p=\{x\in\mathbb Q_p:v_p(x)\ge0\} =\{x:|x|_p\le1\}. \] Algebraically, \[ \mathbb Z_p\cong\varprojlim_n\mathbb Z/p^n\mathbb Z, \] and every \(x\in\mathbb Z_p\) has a unique expansion \[ x=\sum_{n\ge0}a_np^n,\qquad a_n\in\{0,1,\dots,p-1\}. \] Finally, \[ \mathbb Q_p=\operatorname{Frac}(\mathbb Z_p)=\mathbb Z_p[1/p], \] so every nonzero \(p\)-adic number has a Laurent expansion with only finitely many negative powers of \(p\).