Characterization of Integer has Square Root in P-adic Integers
Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p \ne 2$.
Let $a \in Z$ be an integer such that $p \nmid a$.
Then:
- $\exists x \in \Z_p : x^2 = a$
- $a$ is a quadratic residue of $p$.
That is, an integer $a$ not divisible by $p$ has a square root in $\Z_p$ ($p \ne 2$) if and only if $a$ is a quadratic residue of $p$.
Proof
Let $F \in \Z[X]$ be the polynomial:
- $\map F X = X^2 - a$
By definition of formal derivative the formal derivative of $F$ is:
- $\map {F'} X = 2X$
Necessary Condition
Let there exist $x$ such that $x^2 = a$.
By definition of root of polynomial:
- $\map F X$ has a root in $\Z_p$.
From Characterization of Integer Polynomial has Root in P-adic Integers:
- there exists an integer sequence $\sequence {a_n}$ such that:
- $(1) \quad \ds \lim_{n \mathop \to \infty} {a_n} = a$
- $(2) \quad \map F {a_n} \equiv 0 \mod {p^{n + 1} }$
We have:
- $a_0^2 - a \equiv 0 \pmod p$
That is:
- $a_0^2 \equiv a \pmod p$
Hence by definition:
- $a$ is a quadratic residue of $p$.
$\Box$
Sufficient Condition
Let $a$ be a quadratic residue of $p$.
By definition of quadratic residue of $p$:
- $\exists b \in \Z : a \equiv b^2 \pmod p$
Then:
- $\map F b = b^2 - a \equiv 0 \pmod p$
and
- $\map {F'} b = 2b$
By hypothesis:
- $p \nmid 2$
and
- $p \nmid b^2$
From the contrapositive statement of Divisor Divides Multiple:
- $p \nmid b$
From the contrapositive statement of Euclid's Lemma for Prime Divisors:
- $p \nmid 2b$
Hence:
- $\map {F'} b = 2b \not\equiv 0 \pmod p$
From Congruence Modulo Equivalence for Integers in P-adic Integers:
- $\map F b \equiv 0 \pmod {p\Z}$
and
- $\map {F'} b \not\equiv 0 \pmod {p\Z}$
From Hensel's Lemma for P-adic Integers:
- $\exists x \in \Z_p : \map F x = 0$
That is:
- $\exists x \in \Z_p : x^2 = a$
$\blacksquare$
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.7$ Hensel's Lemma and Congruences: Theorem $1.43$