Brahmagupta's identity

In algebra, Brahmagupta's identity says that, for given ${\displaystyle n}$, the product of two numbers of the form ${\displaystyle a^{2}+nb^{2}}$ is itself a number of that form. In other words, the set of such numbers is closed under multiplication. Specifically:

${\displaystyle {\begin{aligned}\left(a^{2}+nb^{2}\right)\left(c^{2}+nd^{2}\right)&{}=\left(ac-nbd\right)^{2}+n\left(ad+bc\right)^{2}&&&(1)\\&{}=\left(ac+nbd\right)^{2}+n\left(ad-bc\right)^{2},&&&(2)\end{aligned}}}$

Both (1) and (2) can be verified by expanding each side of the equation. Also, (2) can be obtained from (1), or (1) from (2), by changing b to −b.

This identity holds in both the ring of integers and the ring of rational numbers, and more generally in any commutative ring.