Can each of the following be a character of a finite group $G$?
$(i) \ \ (2,0,\frac{1}{2},\frac{1}{2},0)$
$(ii) \ \ (3,-1,0,4,0)$
$(iii) \ \ (2,2,2,2)$
$(iv) \ \ (1,0,-1,0)$
I think that the first one cannot be a group as there are $1/2$'s and this contradicts some theorem about characters being algebraic integers. I can see no reason why the others cant be characters of finite groups. Is this correct?
How would we give an example of a finite group with the given character? Why should I be considering?