Here is a theorem from field theory.
Let E be an entension field of field F.
Let 'a' in E be algebraic over F.
Then consider the simple extension F(a) over F.
If 'b' is in F(a) then deg(b,F) divides deg(a,F).
You are working with simple extension field Q(sqrt(2)) over Q.
Trivially, sqrt(2) is in Q(sqrt(2))
We assume it square root is in Q(sqrt(2)).
That is, sqrt(sqrt(2))=4throot(2).
Now, irr(4throot(2),Q)=x^4-2 (use Eisenstein Criterion ).
And 4 does not divide 2.
Thus, Q(sqrt(2)) cannot contain its square roots.