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In any commutative unital ring, if x {\displaystyle x} and y {\displaystyle y} can each be written as a sum of two squares, so can x y {\displaystyle xy} .
More concretely, if x = a 2 + b 2 {\displaystyle x=a^{2}+b^{2}} and y = c 2 + d 2 {\displaystyle y=c^{2}+d^{2}} , then x y = ( a c + b d ) 2 + ( a d − b c ) 2 {\displaystyle xy=(ac+bd)^{2}+(ad-bc)^{2}} . In other words, for all a , b , c , d {\displaystyle a,b,c,d} in a commutative ring:
( a 2 + b 2 ) ( c 2 + d 2 ) = ( a c + b d ) 2 + ( a d − b c ) 2 {\displaystyle \!(a^{2}+b^{2})(c^{2}+d^{2})=(ac+bd)^{2}+(ad-bc)^{2}}
and 
can each be written as a sum of two squares, so can 
.
and 
, then 
. In other words, for all 
in a commutative ring:
