When you talk about groups $[G:H]$ is the number of H-cosets in G. My book has recently started using this notation with fields, and I'm not sure what it means.
My first thought was that you could treat the field as a group under addition. But it seems like it's saying $[Q(\sqrt{2}):Q]=2$ and I'm not sure why this would be. I think the elements of $Q(\sqrt{2})$ are $a+b\sqrt{2}$ and if $b_1 \not=b_2$ then it doesn't seem like $a_1+b_1\sqrt{2} + Q = a_2 + b_2\sqrt{2} + Q$, implying an infinite number of cosets.
When I try using considering the field minus zero as a group under multiplication I also seem to get an infinite number of cosets.
What am I doing wrong?