Cohl Furey sounds like a goddamn new Marvel character, but actually she's a mathematical physicist. Can that really be her name? As a novelist, I feel reality is cheating by being so fucking dramatic:https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/ …
Continuity isn't the right concept here, finite fields have the discrete topology by default. And every finite field is an extension of Z/p for some prime p -- such a field is said to have 'characteristic p' whereas R, C etc. are 'characteristic 0'.
-
-
Generally a structure with addition, subtraction, and multiplication is called a 'ring'; a ring that's also a vector space over a field F is called 'an algebra over the base field F', where F can be R or a finite field or whatever.
-
The original assertion condenses to 'the only finite-dimensional, possibly nonassociative, division algebras over the real numbers are R, C, H, O' Once you get down into the weeds you have to be super careful about what you say, it gets exhausting sometimes.
-
If algebras are rings, and rings have to be associative in their multiplication, then what do you mean by a “possibly non-associative division algebra”?
-
I mean "we're weakening this specific requirement", ie 'algebraic structure that is a division algebra except we give up associativity'.
-
It's a theme: R -> C loses linear ordering C -> H loses commutativity H -> O loses associativity
-
Which of course is a general phenomenon -- drop an axiom and the potential structures explode. Part of what makes this progression interesting to a lot of people is that it *doesn't* explode, you get just one new structure at each step.
End of conversation
New conversation -
-
-
I meant finite-dimensional rather than finite. Yeah, finite fields clearly can’t be continuous, why did I even type that?
Thanks. Twitter will use this to make your timeline better. UndoUndo
-
Loading seems to be taking a while.
Twitter may be over capacity or experiencing a momentary hiccup. Try again or visit Twitter Status for more information.
. Banned in Sweden. SubGenius, Zhuangist, white-hat troll. Defrocked mathematician. Brain problems.