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seen Dec 19 at 14:45

Dec
19
comment Describe the Galois Group of a field extension
thank you so much for your help. Mainly I'm trying to understand if there's a criterion under which the Galois group of a polynomial is the entire symmetric group, and was postulating that (perhaps) you were suggesting an irreducible polynomial will be the whole symmetric group while a reducible one will have some permutation that doesn't work.... also, on that subject, these permutations must also respect the field operations -- is that a material constraint typically, or do all perms of roots work by def'n of the extension? Confused there too :(
Dec
18
comment Describe the Galois Group of a field extension
I may have oversimplified, I mainly meant that it's the product of two quadratics, so to me doesn't that mean it's reducible? I thought $P$ is reducible if there is some $P_1, P_2$ such that $P = P_1 P_2$.
Dec
18
comment Describe the Galois Group of a field extension
I am having the same problem as OP, but I particularly get lost in degrees > 2. I understand your example of $(x^2 - 3)(x^2 - 2)$ but that polynomial is clearly reducible, and I thought it only made sense to apply this theory to irreducible polynomials? Is the reducibility the thing that makes it so the Galois group isn't all of $S_n$? TIA!
Dec
10
comment Proving irreducibility of polynomial over various fields
Ah, I see now. I get the splitting field part at least... the degree of the extension must be the degree of the polynomial, so F_81 splits P. But I'm also supposed to prove (ir)reducibility over F_27 and F_9. I was thinking F_27 isn't a subfield of F_81 (the splitting field) but does that really say anything about reducibility? And for F_9, since it is a subfield, how do I show it irreducible therein, since I can't directly compute as in the case of F_3?
Dec
9
comment Proving irreducibility of polynomial over various fields
Thank you so much for this great answer. I'm having two difficulties understanding, TIA! 1. What about Q? and 2. given the result for F_3, what can I say about F_{3^n} and what theorem / machinery lets me do that?
Dec
9
asked Proving irreducibility of polynomial over various fields
Nov
17
asked Showing two field extensions (of Q) are isomorphic using primitive elements, and why every element is primitive
Oct
30
comment how to transform word problem to an equation ?? is there any trick for this?
I don't really understand your question, but if you're doing this for precalc I might suggest that this is a guess-and-check problem. When I looked at this I thought perhaps it could be solved using some deep result of number theory or ring theory but I doubt that's the idea here. Just start at 2, count up, and crack the spies' code that way.
Oct
30
comment Minitab help/ getting standard deviation from Poisson with mean
I think you either are missing some info in your question or perhaps your assignment is simply to write a computer program and give the sample stdev. To do this just write a computer program that draws from a Poisson distribution 1000 times and calculate the sample standard deviation using the normal formula for that... it's an easier exercise than it sounds I think
Oct
21
awarded  Commentator
Oct
21
comment Splitting a short exact sequence of orthogonal groups
Could you elaborate? I thought i'd seen this before.... couldn't you map $O_n(\mathbb{R})$ to its determinant, and for $SO_n(\mathbb{R})$ use the (trivial) inclusion map?
Oct
21
comment Stochastic processes on group-valued variables
Oh, Robert Adler's work was what I was looking for. So the posted solution was precisely what I needed. I had some very interesting applications for that. But at the end of the day I found some more interesting "applied algebraic topology" from a fellow at Stanford (google "Topology and Data" -- very interesting) and focused on that for a while. The problem domain was of course different but the excursion set solution entirely solved the first problem, so I needed something new. :)
Oct
21
comment Stochastic processes on group-valued variables
I had quite a good time with all that but the subject matter is different, it's still always considering stochastic processes on $\mathbb{R}^n$. This time, my interest is in stochastic processes taking values from a group, as opposed to values in real n-space.
Oct
20
asked Stochastic processes on group-valued variables
May
12
awarded  Self-Learner
Jun
6
awarded  Critic
Jun
6
comment $10=c+d$ and $c$ is one more than $d$.
As above this can be solved immediately.
Sep
23
comment Applications of algebra and/or topology to stochastic (or Markov) processes
It's so annoying, goddam Windows crashed and I had to reboot so I lost all my open tabs in Chrome. Now I can't find the single best piece of math I've ever seen. It even had a discussion about how the reason most people haven't heard of the application is because most mathematicians aren't interested in both statistics and topology. Fascinating.
Sep
22
asked Applications of algebra and/or topology to stochastic (or Markov) processes
Sep
16
awarded  Teacher