$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) \cong \Gal(K/(K \cap L))$$ and $$[KL : F] = \frac{[K : F][L : F]}{[K \cap L : F]}$$

This immediately reminds me of the second isomorphism theorems. For groups: If $H \le G$ and $N \trianglelefteq G$, then: $$ HN/N \cong H/(H \cap N) $$ and $$ |HN| = \frac{|H||N|}{|H \cap N|}, \textrm{ equivalently } |G : HN| = \frac{|G : H||G : N|}{|G : H \cap N|} $$

For rings: if $S$ is a subring and $I$ is an ideal of $R$, then: $$ (S + I)/I \cong S/(S \cap I) $$ and $$ |S + I| = \frac{|S||I|}{|S \cap I|}, \textrm{ equivalently } |R : S + I| = \frac{|R : S||R : I|}{|R : S \cap I|} $$

If $K$ is replaced with $H$ (or $S$), $L$ with $N$ (or $I$), and degree with index, these become the same. I can see the connection between the field and group versions with the Fundamental Theorem of Galois Theory, and the group and ring versions by just verifying that multiplication still checks out. But it feels like this is a statement about algebraic structures in general.

Is there a way of showing this holds for certain kinds of structures? It feels like a problem for homological algebra or category theory, but I don't know enough about either to tackle it myself.

EDIT: More examples. The second isomorphism theorem for modules fits this mold, and it's proved pretty much the same as for rings.

In particular, this works for vector spaces. But the "size function" doesn't have to be the index (which is not helpful for $\mathbb{R}$-spaces, for example), it can also be dimension (which is what's going on with the Galois groups). The Subspace Sum-Intersection theorem seems to fit this mold as well: for two subspaces $S$ and $T$ of $V$, $\dim (S + T) = \frac{\dim S + \dim T}{\dim (S \cap T)}$.

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    $\begingroup$ relevant? en.wikipedia.org/wiki/Isomorphism_theorem#General $\endgroup$ May 1, 2014 at 3:47
  • $\begingroup$ This definitely fits what I was looking for, for groups and rings (and I think vector spaces, but I'm not sure yet), but I'm having trouble applying this to Galois groups. $\endgroup$ May 1, 2014 at 4:36
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    $\begingroup$ Isn't the second isomorphism theorem true in every abelian category? math.stackexchange.com/q/2415957/263430 as a reference. I think its nicer to see this with mitchell's embedding theorem which roughly states that there's a correspondence (equivalence even) between every abelian category and R-mod for some R. $\endgroup$ Sep 26, 2017 at 22:03
  • $\begingroup$ What is the relevant abelian category in field extensions of a given field? $\endgroup$ Apr 20, 2018 at 1:25
  • $\begingroup$ I would look into lattice theory. In all examples, we're dealing with joins and meets of certain subobjects, and the statement is that certain edges in the diagram are similar ("parallel" or "of same length"). Cf. math.stackexchange.com/q/1738334/96384, math.stackexchange.com/q/2227241/96384, and there is something in the answers linked to by @BenjaminGadoua that might work under less restrictive assumptions than abelianness of the category. Certainly it will not always work, but it would be interesting to see what exactly-abstractly makes it work sometimes. $\endgroup$ Dec 2, 2020 at 4:34


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