---
tag: 00DW862T78
kind: statement
role: corollary
name: −1 is a square modulo p if and only if p is not 3 modulo 4
msc: [11A15]
uses: [0PHBS78HKT, 0SN0MAEKYX]
sources:
  - url: https://en.wikipedia.org/wiki/Quadratic_reciprocity
    locator: q = ±1 and the first supplement
---

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
There is an integer $x$ with $x^2 \equiv -1 \pmod p$ if and only if $p \not\equiv 3 \pmod 4$.

## Remarks

For $p = 2$, take $x = 1$.
For odd $p$, the statement is the [[first supplement to quadratic reciprocity|0PHBS78HKT]].
For $p \equiv 1 \pmod 4$, an explicit solution is $x = \left(\frac{p-1}{2}\right)!$, by [[Wilson's theorem|03KGGKTB2E]].
The statement is a step in the proof of [[Fermat's theorem on sums of two squares|043B3ENSTY]].

## Explanations

### L1 human

The primes $2$, $5$, $13$ and $17$ have a square root of $-1$: for example $5^2 = 25 \equiv -1 \pmod{13}$.
The primes $3$, $7$ and $11$ have none.
The rule is that $p$ must not leave remainder $3$ on division by $4$.