Linearity of the map Field Trace.

Question

!I want to prove that this map is well-defined. I know that the image of 'a' is sum of all Galois conjugates of 'a' i.e. the roots of minimal polynomial 'a' over Fq. I can prove the linearty of this map but my question is how this sum is an element of Fq. Please help me I have stucked here.