Let $F$ be a field and let $a,b,c,d\in F\Rightarrow (a^2+b^2)(c^2+d^2)$ can be written as $x^2+y^2$ for some $x,y\in F$?

Question

Let $F$ be a field and let $G_2$ be the subset of $F$ consisting of all elements which can be written as a sum of $2$ squares of elements of $F$. Is the product of two elements of $G_2$ again an element of $G_2$?

I really dont think that this is true but I have dont found an example that fails.

Some idea of why this is false or true?

This is an excercise from the book: The Linear Algebra a Beginning Graduate Student Should Know; Golan

Answer 1

Yes. If $x\in G_2$ and $y\in G_2$, say $x = a^2 + b^2$ and $y = c^2 + d^2$, then $$xy = (ac - bd)^2 + (ad + bc)^2$$ and $ac - bd, ad + bc\in F$. So $xy\in G_2$.