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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?

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I mentioned this notation in an answer to another question (of yours, in fact). – Dylan Moreland Sep 7 '11 at 15:34
For field extensions, it's the dimension of $G$ as a vector space over $H$. – Henning Makholm Sep 7 '11 at 15:35
@Dylan: wow, that's embarrassing. Usually evidence of my bad memory is better hidden. – Xodarap Sep 7 '11 at 15:56
up vote 1 down vote accepted

For fields $L$ and $K$ with $K \subset L$, $[L : K]$ is the dimension of $L$, considered as a vector space over $K$. So $[\mathbb{Q}[\sqrt{2}]:\mathbb{Q}]=2$, because everything in $\mathbb{Q}[\sqrt{2}]$ can be expressed uniquely as $a+b \sqrt{2}$ with $a, b$ in $\mathbb{Q}$.

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As a comment, I'd like to point out that using the same notation for different notions (degree of a field extension and index of a subgroup) is useful, the Galois correspondence theorem being its prime evidence. – Bruno Stonek Sep 7 '11 at 15:45

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