Same symbol, different meanings? Subgroup index & field extension

[nonequilibrium]

Hello,

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

But we can also see F and E as the groups (F,+) and (E,+), and then the same symbol denotes the subgroup 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 not equal (doesn't mean there can't be some meaningful connection, however), because for example as fields $[ \mathbb C : \mathbb R ] = 2$ while as groups $[ \mathbb C : \mathbb R ] = 2^\aleph_0$ (anyway I think so, because $\mathbb C = \cup_{r \in \mathbb R} (r.i + \mathbb R)$)

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

 

[micromass]

while as groups $[ \mathbb C : \mathbb R ] = \aleph_0$

Are you certain of this?? Doesn't this need to be

$$[\mathbb{C}:\mathbb{R}]=2^{\aleph_0}$$

since the reals have cardinality $2^{\aleph_0}$.

In general if F is a field extension of K, then we can write $F=K^n$ as vector space (and where n is not necessarily finite).

So, as groups, we have

$$[F:K]=|F/K|=|K^n/K|=|K^{n-1}|$$

while as fields (or vector spaces), we have

$$[F:K]=\dim_K{F^n}=n$$

In general, it seems obvious that [F:K] as fields is always smaller than [F:K] as groups. But I cannot quite see any other interesting relation.

 

[spamiam]

Well, there are several similarities. For a group, [itex] [G] [/itex] is the number of distinct left cosets of H in G, or equivalently, the number of elements in a set of coset representatives for left cosets of H in G. Similarly, $[F:E]$ is the number of elements in a basis for F as a vector space over E. Other results hold in both cases as well. For instance, if $F \subseteq K \subseteq L$ are fields or groups, then
$$[L:F] = [L:K][K:F] \; .$$
I think there are other results that are true for both interpretations of the symbol [F:E], often involving divisibility.

Also micromass is right: as a group under addition, $\mathbb{C} \cong \mathbb{R}^2$ so $[\mathbb{C}: \mathbb{R}] = |\mathbb{R}| = 2^{\aleph_0}$.

 

[nonequilibrium]

Indeed the cardinality was a typo (fixed it in my OP)

thanks for the rest

 

[blue_raver22]

Hello,

Thank you for bringing up this interesting topic. I would say that there is indeed a relation between these two cases, although they may seem to be unrelated at first glance.

In the context of fields and field extensions, the symbol [E:F] represents the degree of the field extension, which is the dimension of the vector field E across the field F. This is a measure of how much the field E extends the field F.

On the other hand, in the context of groups, the same symbol [E:F] represents the subgroup index, which is the number of cosets of the subgroup E in the group F. This is a measure of how many times the subgroup E fits into the group F.

While these two measures may seem different, they are actually related in a meaningful way. In fact, the degree of a field extension [E:F] is equal to the index of the subgroup [F:E]. This can be seen through Galois theory, which studies the relationship between fields and groups.

So while the two cases may use the same symbol, they are not completely unrelated. There is a deeper connection between them that can be explored through mathematical concepts such as Galois theory.

I hope this helps clarify the relationship between these two cases. Thank you for bringing this up and for your interest in this topic.