Alg3.4.9

consider $x 2 = a, y 2 = b$

we want to factor $a 2 + a b + b 2$

and this we cannot factor because if we

calcualte the discriminant of the equation(in a) we get

$Δ = b 2 - 4 b 2 = - 3 b 2 < 0$ so the

equation has no roots in $\mathbb{R}$

so it is obvious it cannot be decomposed as

$c (a - a 1)(a - a 2)$ where

c is a constant and $a 1, a 2$ are depending on

$b$ .

the same procedure is taken out on equation with respect

to b.

the conclusion follows