Exercise
Let m, n be positive integers. By Q + Q
m + Qn
we denote the set of all linear combinations
a + bm + cn
for a, b, c 
Q.
The subset Q + Qm + Qn
of C is a field if and only if one of the conditions below is satisfied,
Which?
m and n are squares.
mn is a square.
m or n is a square.
This is a
sufficient condition (for then the set is just Q), but not a
necessary one.
Correct.
No. For instance,
if m = n is a nonsquare, then
Q+Qm+Qn = Q+Qm is a field.
m+Qn = Q+Qm is a field.