Each Subfield of the field of Complex Numbers contains every rational number

Some Linear Algebra stuff after a long time. Hope to continue the steam if possible.

Problem : Prove that each subfield of the field of complex numbers contains every rational number

Proof :
Let C be the field of complex numbers and let F be any subfield of C
Now, let p/q is one such rational number that does not belong to F
Therefore, -p/q also does not belong to F

Now, let x belongs to F.

Since x belongs to F and p/q does not, hence (x+p/q) does not belong to F
Hence, (x+p/q) + (-p/q) also does not belong to F
=> x + (p/q +(-p/q)) does not belong to F
=> x + 0 does not belong to F
=> x does not belong to F
which is a contradiction.

Hence Proved.