Odd p-Adic Squares Are Detected Modulo p

For odd p, a p-adic number is a square when its valuation is even and its unit residue is a quadratic residue. This is a compact note, but the quantifiers and hypotheses stay on the page.

Notation

The valuation $v_p(n)$ counts factors of $p$, and the metric $|x|_p=p^{-v_p(x)}$ reverses the usual sense of size. Hensel lifting turns approximate roots modulo $p$ into exact $p$-adic roots.

$$ x=p^nu,\qquad u\in\mathbf Z_p^\times $$

I keep the defining relation $\mathsf D$ above the derived relation $\mathsf C$. This exposes whether cancellation used $x\ne0$ and whether the conclusion is canonical.

$$ x\in(\mathbf Q_p^\times)^2\Longleftrightarrow n\equiv0\pmod2\ \text{and}\ \left(\frac{\bar u}{p}\right)=1 $$

Stress the formula

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ \mathbf Q_p^\times/(\mathbf Q_p^\times)^2\cong\{1,\ u,\ p,\ up\}\qquad(p\ \text{odd},\ u\text{ a nonsquare unit}) $$

Interpretation

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad x=p^nu,\qquad u\in\mathbf Z_p^\times,\\[5pt] \mathsf{C}\;&:\quad x\in(\mathbf Q_p^\times)^2\Longleftrightarrow n\equiv0\pmod2\ \text{and}\ \left(\frac{\bar u}{p}\right)=1. \end{aligned} $$

Limit of the argument

A congruence root lifts uniquely only in the simple-root case. Multiple roots need stronger inequalities and may split, disappear, or lift nonuniquely.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] x\in(\mathbf Q_p^\times)^2\Longleftrightarrow n\equiv0\pmod2\ \text{and}\ \left(\frac{\bar u}{p}\right)=1 \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Tue 08 October 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.

Odd p-Adic Squares Are Detected Modulo p by Vicfred

Notation

Stress the formula

Interpretation

Limit of the argument