On expressing elements as a sum of squares, where one square is restricted to a subfield

Let q be a prime power and fix a∈Fq2∖Fq. In this note it is proved that, provided q>5, the setSa={a−α2:α∈Fq⁎} contains both a square and a non-square of Fq2⁎. In particular, every a∈Fq2∖Fq can be written as a sum of a square of Fq2 and a square of Fq.