AdicSqrt

A square root of a AdicNumeral in $\mathbb{Q}_p$, found by Hensel lifting — defined only for prime bases, and only when one exists.

AdicSqrt(x, prec?)the root, to prec digits (default 20)

Typed signatures

Type	Package	Applies
(value, integer?) -> value	numerals	always

Domain: Numeral systems

Details

• Needs an even valuation, and (for odd $p$) a unit part that is a quadratic residue mod $p$; for $p=2$ the unit part must be $1 \bmod 8$

Examples

Basic #

$\sqrt{2} \in \mathbb{Z}_7$, to 6 digits (2 is a quadratic residue mod 7)

live ↗

In$\mathrm{AdicSqrt}(\mathrm{AdicNumeral}(7, 2), 6)$Out$\mathrm{AdicNumeral}(7, 38\,181, 6)$

Scope #

$-7 \equiv 1 \pmod 8$, so it has a square root in $\mathbb{Z}_2$

live ↗

In$\mathrm{AdicSqrt}(\mathrm{AdicNumeral}(2, -7), 10)$Out$\mathrm{AdicNumeral}(2, 181, 10)$

The digits of $\sqrt{2}$, as AdicExpansion writes them

live ↗

In$\mathrm{AdicExpansion}(\mathrm{AdicSqrt}(\mathrm{AdicNumeral}(7, 2), 6))$Out$\text{…216213 + O(7\textasciicircum{}6)}$

About

Examples	3

See also: AdicNumeral, HenselLift