From "Non Local Aspects of Quantum Phases" by J. ANANDAN, also noticing:
More generally I'd say that a differential structure with a tangent bundle is almost always assumed in physics (both classical and quantum, and e.g. including here even Penrose spinor bundles) and I can hardly imagine a...
It sounds like Michael Atiyah says the opposite:
The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics.
And well, regarding the degree, the fact is simply that if P is reducible, then it has a factor of degree less or equal than n over 2: therefore, as per the correspondence you have pointed out, ## \exists ## a stem field of that degree and we can take it as the extension to prove the corollary...
No, I was wrong, I'm reading again the previous lesson.
Yes, you are right about it, they mean a minimal polynomial in K for the root in the extension L.
In my example the minimal polynomial in Q for the root ##\sqrt 2## in R would be P itself ##x^2-2##, which is irreducible in Q.
Even so, it...
The minimal polynomial of a root of a monic polynomial P over a field K must divide P and it is unique and the definition is also in wikipedia... In the mentioned course there was a lesson in the previous week specifically about the definition of Minimal polynomial (that is unitary, etc...).
I'm...
Ops... but now I see that the degree of a transcendental extension is infinite so the fact that the extension is algebraic could be deduced (but only from the constraint on the degree of the extension) without assuming it...
To be more clear, the full corollary one is:
P is irreducible over K if and only if it does not have roots in extensions L of K of degree less or equal than n/2 where n is the degree of P.
So, if I take the direction starting from "P has roots in an extension L of K...", it follows what I have...
The point is that the corallary says
if P has a solution ##\alpha## in L, where L is an extension of K...
which means that the solution and the minimal polynomial are in L, even if P is over K.
My example is the obvious ##x^2+1## in Q which is divided by ##x+i## and ##x-i## in the extension R
Well, as far as I can understand it, let say that K is Q of rational and L is R of real numbers, theoretically, without other assumptions more restrictive (and if I am not wrong). Notice that in this case the extension would be transcendental instead of algebraic.
The point is that the m(x) in...
I believe that the above holds iff the extension ##\mathbf L## is algebraic over ##\mathbf K##.
That would make sense to me and hopefully answers my question.
I was following the week 2 of Ekaterina Amerik introducing Galois theory but this (IMO important) detail is missing in her explanation...
I'm not sure the following passage is so trivial as it was supposed to be: I mean, what does exactly prove it? That's my question.
The step is the following:
if ##P## has a root ##\alpha## in ##\mathbf L## - an extension of ##\mathbf K## of degree <= ##\frac n 2## where n is the degree of ##P##...
Indeed my doubt is somehow reinforced from what I read in "Static replica approach to critical correlations in glassy systems" (same authors, among which again Pierfrancesco Urbani and this year's Nobel Prize, Giorgio Parisi) ref. 12A540-22 paragraph "C. Expression of λ in HNC", page 23, where...
I'm reading the https://www.phys.uniroma1.it/fisica/sites/default/files/DOTT_FISICA/MENU/03DOTTORANDI/TesiFin26/Urbani.pdf at paragrph 4.6.2 "The interaction term".
They write a right hand side:
< f(na,nb) f(nc,nd) f(ne,nf) >
and they want to use a symmetry, for example they assume that...
Ok, maybe I took a too trivial example just to start and indeed you're right, here's more a matter of algebra than topology, in fact my goal is to understand Lie algebra extensions more in general.
On the opposite side of complexity, I would rephrase my question with the example of Virasoro...
In conclusion, I wanted also to ask a more philosophical question, an explanation without all the details but high level: what is the fact of being also a central extension adding on top of a double cover? My understanding is that it is adding a structure of morphisms, maybe a tensor product...