A \(p\)-adic metric declares two rational numbers close when their difference is divisible by a high power of \(p\). Completing \(\mathbb Q\) for this metric yields \(\mathbb Q_p\), where congruences at all powers of \(p\) become one analytic object.
Valuations and ultrametrics
Definition. For \(x\in\mathbb Q^\times\), write \(x=p^ku/v\) with \(p\nmid uv\). Set \[ v_p(x)=k,\qquad |x|_p=p^{-v_p(x)},\qquad |0|_p=0. \]
Proposition. \[ |xy|_p=|x|_p|y|_p,\qquad |x+y|_p\le\max(|x|_p,|y|_p). \] If \(|x|_p\ne|y|_p\), then equality holds in the second formula.
Proof. Additivity of \(v_p\) under multiplication proves the first identity. For addition, \[ v_p(x+y)\ge\min(v_p(x),v_p(y)), \] because the smaller common power of \(p\) factors out. If the two valuations differ, the remaining sum is a unit modulo \(p\), giving equality.
The strong triangle inequality makes every triangle isosceles with the two long sides equal. Every point of a ball is a center, and intersecting balls are nested.
Completion and expansions
Definition. The field \(\mathbb Q_p\) is the completion of \(\mathbb Q\) under \(|\cdot|_p\). Its ring of integers and maximal ideal are \[ \mathbb Z_p=\{x:|x|_p\le1\},\qquad p\mathbb Z_p=\{x:|x|_p<1\}. \]
Theorem. Every \(x\in\mathbb Q_p^\times\) has a unique expansion \[ x=\sum_{n=k}^{\infty}a_np^n,\qquad a_n\in\{0,\ldots,p-1\},\quad a_k\ne0. \] It belongs to \(\mathbb Z_p\) exactly when \(k\ge0\).
Convergence means that the omitted tail is divisible by increasingly high powers of \(p\). For example, in every \(\mathbb Q_p\), \[ -1=(p-1)+(p-1)p+(p-1)p^2+\cdots. \]
Compactness and inverse limits
Theorem. \[ \mathbb Z_p\cong\varprojlim_n\mathbb Z/p^n\mathbb Z. \] Consequently \(\mathbb Z_p\) is compact, totally disconnected, and complete.
Proof. A \(p\)-adic integer determines its compatible residues modulo every \(p^n\). Conversely, compatible residues determine successive digits and hence a Cauchy sequence. The inverse limit is a closed subset of the compact product \(\prod_n\mathbb Z/p^n\mathbb Z\), proving compactness.
Series and local fields
Proposition. In a complete non-Archimedean field, \[ \sum_{n=0}^{\infty}a_n \quad\text{converges if and only if}\quad a_n\to0. \]
Proof. Necessity is universal. For sufficiency, if \(m>n\), \[ \left|\sum_{k=n+1}^ma_k\right| \le\max_{n<k\le m}|a_k|\to0, \] so the partial sums are Cauchy.
Finite extensions of \(\mathbb Q_p\) are local fields. Their valuation rings are discrete valuation rings, and their ramification and residue degree are the local counterparts of prime-ideal factorization in number fields.