Smallest subfields containing Z and Z[sqrt2]

[gotmilk04]

Smallest subfields containing Z and Z[sqrt2]

Homework Statement

What are the smallest subfields of R containing Z and Z[$$\sqrt{2}$$]?

Homework Equations

Z= {a+ib|a,b$$\in Z$$}
Z[$$\sqrt{2}$$]={a+b$$\sqrt{2}$$|a,b$$\in Z$$}

The Attempt at a Solution

Z$$\subset$$Q and Z[$$\sqrt{2}$$]$$\subset$$Q[$$\sqrt{2}$$]
so to get the subfield that contains both of them, would it be Q[i,$$\sqrt{2}$$]?

 

[HallsofIvy]

Yes.