Brahmagupta-Fibonacci two-square identity

Statement

In any commutative unital ring, if $x$ and $y$ can each be written as a sum of two squares, so can $xy$.

More concretely, if $x=a^2+b^2$ and $y=c^2+d^2$, then $xy=(ac+bd{)}^2+(ad-bc{)}^2$. In other words, for all $a,b,c,d$ in a commutative ring:

$(a^2+b^2)(c^2+d^2)=(ac+bd{)}^2+(ad-bc{)}^2$

Related facts

Applications

• Positive integer is a sum of two squares iff it has no prime divisor that is 3 mod 4 with odd multiplicity