Let K be an algebraic number field, and L be the Hilbert class field with the property that every irreducible in K factors uniquely (up to units) into irreducibles in L. Then K and L are both finite-dimensional vector spaces over Q, and the class number is the ratio of the dimensions. Now that I think about it, “index” is already a standard term for things like this. Jim On Sun, May 24, 2020 at 7:01 PM Dan Asimov <dasimov@earthlink.net> wrote:
Why not tell us which vector spaces the class number is a ratio of.
—Dan
----- Here’s an example of a context in which such a ratio is relevant (unless my understanding of Hilbert class fields is wrong, in which case someone please set me straight):
The class number h of an algebraic number field K can be understood as a ratio of the dimensions of two vector spaces, though normally we would express this in terms of fields rather than vector spaces, saying that the Hilbert class field of K is a degree-h extension of K.
Could we call K a “codegree-h” “intension” of the Hilbert class field of K?
Jim Propp
On Sun, May 24, 2020 at 3:33 PM James Propp <jamespropp@gmail.com> wrote:
If W is a subspace of V, is there a name for dim V / dim W? And more generally for the ratio of the dimensions of two things, one of which sits inside the other?
(This question is inspired by my misrecollection of the nature of knotting in higher dimensions, which is the same as Cris Moore‘s: we both recalled that 2 was involved, but we thought it was relevant as an approximate ratio, rather than as a difference!)
I could defend the term “codegree”, but I’m hoping someone else already coined an obscure but uncontroversial word for dimension-ratios.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun