Definition:Biquadratic Residue

Definition

Let $p$ be an odd prime.

Let $a \in \Z$ be an integer such that $a \not \equiv 0 \pmod p$.

Then $a$ is a biquadratic residue of $p$ if and only if $x^4 \equiv a \pmod p$ has a solution.

That is, if and only if:

$\exists x \in \Z: x^4 \equiv a \pmod p$

Also known as

A biquadratic residue is also known as a quartic residue.

Also see

• Definition:Quadratic Residue

Historical Note

The concept of a biquadratic residue was investigated by Carl Friedrich Gauss in his papers of $1828$ and $1832$.