You're right about aleph_1, my mistake, though it doesn't really matter here.
About the picks: I think you got me wrong. what we are picking is the next digit in the binary expension of a real number between 0 and 1. As far as I know there is a countable number of digits in the expension...
after rereading, I saw there is a mistake in 2:
I didnt meant to say the time to pick a random number in {0,1} could be un-bounded but, that it won't take some minimal time or formally, for all e>0, I can always pick such a number in less than e seconds.
the physical diffference between a...
Hi,
I am currently working on probability densities and one simple question came to my mind:
Say we lived in an "2^aleph_0 world" which contains 2^aleph_0 mathematicans (dont screem already!). Now each mathematician chooses one real number between 0 and one in a bijective way.
Now we can pick...
I think about it that way:
For n=3 for exemple,
Think at A as a linear transformation on R3: then you can think at A^T A as a convex bilinear transformation on R3xR3 giving you the scalar product of two vectors A(x1) and A(x2)
In particular, when x1 = x2 = x, I think at it as a "lengh"...
for all n define the finite sequence of n+1 elements
1/n^(1/3) , 1/(n*n^(1/3)), ..., 1/(n*n^(1/3))
when ... means n times 1/(n*n^(1/3)).
for all n, put those sequences on a row and look at the sequence a_i you get.
then sum (a_i) converges since for all n sum(a_i) < 1/(n*n^(1/3)) after some...
Ok, I found the solution in a book:
You have to look at the equation X^p - a = 0 mod P^(p*e/(p-1))
when P stands for the prime ideal of your field, p the characteristic and e the absolute ramification index:
It has a solution iff the extension is unramified.
In the case of a more general...
Indeed the correspondence you're talking about is true.
Now, forget about ramification groups: In a first step, I only need to find a simple (or maybe more complicated) criterion for deciding whether the local field defined by x^p - a is ramified or not.
Artin-Shreier theory, in my...
Ok, so I think the problems comes from the fact that reducing the equation x^p-a modulo P does NOT give a defining equation for the coresponding residue field extension.
Then the question arising is how can one determine such a defining equation on residue fields, given the local extension...
Ok, but k2 contains a primitive 3 root of unity, doesn't Kummer theory exactly says that 3-cyclic extensions of k2 are ALL generated by such a polynomial?
I don't care much about k1: the only thing I need to know about it is that it is cyclic unramified of degree 3... I know for sure there is such a field over F from class field theory for exemple
Hi everyone,
This is my first post here, so first of all, Congrats on this website, it seems very instructive and usefull!
I m currently studying p-adic number fields for my master degree and have a problem I can't seem to resolve.
Consider the p-adic local field F=Q_3 and its 2...