If $a$ is a totally positive real cyclotomic numbers, then it is a sum of two squares of real cyclotomic numbers.

it suffices to check that the equation $x^2+ y^2 - a z^2=0$ has solutions in real cyclotomic numbers. It has solutions in a particular real cyclotmic number field $F$ if it has solutions everywhere locally. This equation has solutions locally if and only if the quaternion algebra $(a,-1)$ splits locally.


First take the field $F$ generated by $a$. If we adjoin to $F$ a sufficiently large totally real extension of the field of $2$-power roots of unity (sufficiently large depending on the set of ramified places of this quaternion algebra), producing a field $K$, then every ramified place $v$ of this quaternion algebra will not be totally split in $K$, which because $K/F$ is a Galois extension of degree a power of $2$, implies that the quaternion algebra splits over $K_v$, and so the equation has solutions locally over $K_v$, thus solutions over $K$.