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In algebra, Brahmagupta's identity says that, for given n {\displaystyle n} , the product of two numbers of the form a 2 + n b 2 {\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:

Both and can be verified by expanding each side of the equation. Also, can be obtained from , or from , 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.

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