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Say we have field (F,+,.) and field extension (E,+,.), then thedegree of the field extension(i.e. the dimension of the vector field E across the field F) is given the symbol [itex] [E:F] [/itex].

But we can also see F and E as the groups (F,+) and (E,+), and then the same symbol denotes thesubgroup index(in relation to Lagrange's theorem).

Is there any relation between these two cases, or is it just that the same symbol is used for two totally different things and it's assumed that the context makes clear what is meant?

In any case they're notequal(doesn't mean there can't be some meaningful connection, however), because for example as fields [itex] [ \mathbb C : \mathbb R ] = 2 [/itex] while as groups [itex] [ \mathbb C : \mathbb R ] = 2^\aleph_0 [/itex] (anyway I think so, because [itex]\mathbb C = \cup_{r \in \mathbb R} (r.i + \mathbb R)[/itex])

Then again for the rationals and irrationals they seem to coincide, but that might well be a coincidence.

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# Same symbol, different meanings? Subgroup index & field extension

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