Proof that −1 is a square modulo p if and only if p is not 3 modulo 4

Builds on First supplement to quadratic reciprocity Legendre symbol Quadratic residue Prime number

Proof

Let $p$ be a prime number. If $p = 2$ , then $x = 1$ gives $x^2 = 1 \equiv -1 \pmod 2$ . Also $2 \not\equiv 3 \pmod 4$ . So the statement holds for $p = 2$ .
Let $p$ be odd. Then $p \equiv 1$ or $p \equiv 3 \pmod 4$ .
There is an integer $x$ with $x^2 \equiv -1 \pmod p$ exactly when $-1$ is a quadratic residue modulo $p$ .
Since $p \nmid -1$ , this holds exactly when the Legendre symbol $\left(\frac{-1}{p}\right)$ equals $1$ .
By the first supplement, $\left(\frac{-1}{p}\right) = (-1)^{(p-1)/2}$ . This is $1$ exactly when $(p-1)/2$ is even, that is, when $4 \mid p - 1$ .
So for odd $p$ , a solution exists exactly when $p \equiv 1 \pmod 4$ . By step 2, this means $p \not\equiv 3 \pmod 4$ .

Idea

For odd $p$ , the first supplement gives the answer as the sign $(-1)^{(p-1)/2}$ , which is $+1$ exactly for $p \equiv 1 \pmod 4$ . The prime $2$ is checked by hand.

Reading path

To understand this page from the ground up, read these 77 pages in order. The foundations come first. The path takes the proofs Proof of Bézout's identity, Proof of cancellation in an integral domain, Proof of the generalized Euclid's lemma, Proof of the commutativity of addition by induction, Proof of Fermat's little theorem by rearrangement, Proof of Euclid's lemma, Proof that R[x] is an integral domain iff R is, Proof of the degree of a product over an integral domain, Proof of strong induction from ordinary induction, Proof that the integers modulo a prime form a field, Proof that congruence is compatible with addition, Proof of Euler's criterion by counting quadratic residues, Proof that a polynomial over a field has at most as many roots as its degree, Proof of the remainder theorem, Proof that congruence is compatible with multiplication, Proof of the well-ordering principle from strong induction, Proof of the division algorithm, Proof of the first supplement to quadratic reciprocity from Euler's criterion, Proof of the factor theorem, Proof that congruence is compatible with powers.

Definition by recursion concept
Divisibility definition
Group concept
Mathematical induction concept
Natural numbers object
Peano axioms concept
Congruence modulo n definition
Greatest common divisor definition
Prime number definition
Ring definition
Successor function definition
Quadratic residue definition
Residue class definition
Proof that congruence is compatible with addition proof
Proof that congruence is compatible with multiplication proof
Coprime integers definition
Addition of natural numbers definition
Commutative ring definition
Polynomial ring definition
Ring homomorphism definition
Unit of a ring definition
Zero divisor definition
The successor function is injective statement
Zero is not a successor statement
Congruence is compatible with addition statement
Legendre symbol definition
Congruence is compatible with multiplication statement
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
Divisibility in a commutative ring definition
Degree of a polynomial definition
Polynomial evaluation definition
Field definition
Integral domain definition
Integers modulo n definition
Proof that congruence is compatible with powers proof
Commutativity of addition statement
Leading coefficient definition
Proof of cancellation in an integral domain proof
Proof of strong induction from ordinary induction proof
Root of a polynomial definition
Evaluation homomorphism definition
Congruence is compatible with powers statement
Monic polynomial definition
Strong induction statement
Multiplicity of a root definition
Proof of the degree of a product over an integral domain proof
Cancellation in an integral domain statement
Proof of the remainder theorem proof
Proof of the well-ordering principle from strong induction proof
Division algorithm for polynomials statement
Degree of a product statement
Well-ordering principle statement
Remainder theorem statement
Proof that R[x] is an integral domain iff R is proof
Proof of the division algorithm proof
Proof of the factor theorem proof
R[x] is an integral domain iff R is statement
Division algorithm statement
Factor theorem statement
Proof of Bézout's identity proof
Proof that a polynomial over a field has at most as many roots as its degree proof
Bézout's identity statement
A polynomial over a field has at most as many roots as its degree statement
Proof that the integers modulo a prime form a field proof
Proof of the generalized Euclid's lemma proof
Integers modulo a prime form a field statement
Generalized Euclid's lemma statement
Proof of Euclid's lemma proof
Euclid's lemma statement
Proof of Fermat's little theorem by rearrangement proof
Fermat's little theorem statement
Proof of Euler's criterion by counting quadratic residues proof
Euler's criterion statement
Proof of the first supplement to quadratic reciprocity from Euler's criterion proof
First supplement to quadratic reciprocity statement
Proof that −1 is a square modulo p if and only if p is not 3 modulo 4 proof

Where this leads

6 pages build on this one.

Tag	`DZ5NDRMMGR`
URI	`m2://DZ5NDRMMGR`
Kind	proof
Writing	W0 stub
MSC	11A15
Proves	−1 is a square modulo p if and only if p is not 3 modulo 4
Uses	First supplement to quadratic reciprocity, Legendre symbol, Quadratic residue, Prime number
Changed	4 November 2026

Proof that −1 is a square modulo p if and only if p is not 3 modulo 4

Proof

Idea

Reading path

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