Euler's four-square identity

Statement

As an identity

In any commutative unital ring, if $a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}$ are elements, then:

(a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2}) (b_{1}^{2} + b_{2}^{2} + b_{3}^{2} + b_{4}^{2}) =

(a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3} - a_{4} b_{4})^{2} +

(a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{4} - a_{4} b_{3})^{2} +

(a_{1} b_{3} - a_{2} b_{4} + a_{3} b_{1} + a_{4} b_{2})^{2} +

(a_{1} b_{4} + a_{2} b_{3} - a_{3} b_{2} + a_{4} b_{1})^{2} .

In terms of products two at a time

In any commutative unital ring, if $x$ and $y$ are both elements that can be expressed as a sum of four squares, then $x y$ can also be expressed as a sum of four squares.

In terms of products of arbitrary length

In any commutative unital ring, if $x_{1}, x_{2}, \dots, x_{n}$ are all elements that can be expressed as a sum of four squares, then $x_{1} x_{2} \dots x_{n}$ can also be expressed as a sum of four squares.

Related facts

Similar facts

Applications

Lagrange's four-square theorem