Let $a>0$ be a real cyclotomic number. Is it always possible to solve in cyclotomics the equation $X\overline{X}=a$ ?

Equivalently, one might want to express $a$ as a sum of squares of two real cyclotomics. It is well-known that one square is not always enough.

(If two squares are not enough, then, is there an upper bound?).

Let $a>0$ be a real cyclotomic number. Is it always possible to solve in cyclotomics the equation $X\overline{X}=a$ ?

Equivalently, one might want to express $a$ as a sum of squares of two real cyclotomics. It is well-known that one square is not always enough.

(If two squares are not enough, then, is there an upper bound?).

Edit: $a$ is not only positive, but totally positive (otherwise the answer is No).