  Let K be a totally real Galois number field. C. J. Hillar proved that if f in
Q[x_1,...,x_n] is a sum of m squares in K[x_1,...,x_n], then f is a sum of N(m)
squares in Q[x_1,...,x_n]. Modifying Hillar's proof, we improve the improve the
bound given for N(m), the proof being constructive as well.
