Characterization of Integer has Square Root in P-adic Integers/Necessary Condition
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Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p \ne 2$.
Let $a \in Z$ be an integer such that $p \nmid a$.
Let the exist $x \in \Z_p$ such that $x^2 = a$.
Then:
- $a$ is a quadratic residue of $p$.
That is:
- an integer $a$ not divisible by $p$ has a square root in $\Z_p$ ($p \ne 2$)
- $a$ is a quadratic residue of $p$.
Proof
Let $F \in \Z \sqbrk X$ be the polynomial:
- $\map F X = X^2 - a$
By definition, the formal derivative of $F$ is:
- $\map {F'} X = 2 X$
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 $p$.
$\blacksquare$