[qcoh]

> I'd have to look up what the "/" in "M/L" means in the 5th reply (I didn't)

M/L means that M is a field-extension over L. A concrete example would be C/R (the complex numbers over the reals).

Algebraic number theory, in particular Galois theory, studies field extensions by looking at the group of symmetries: the field automorphisms of the larger field that fix the smaller field. For the concrete example above, the Galois group is a group with two elements: the identity function and the function that maps i to -i and keeps real numbers fixed. It's not a coincidence that the dimension of C as an R-vector space is the same as the size of this group (or that the degree of the polynomial that has i (and -i) as roots has degree 2).

[skulk]

If anyone is wondering why field extensions aren't transitive, you can watch Fields medalist Dr. Richard Borcherds explain it: https://youtu.be/zAqVrdNE81g?t=798