William
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 Mar 24 awarded Tumbleweed Mar 18 comment How can I understand cohomology theories in the context of basic homology theory? Yeah I'm pretty well-versed in graph theory and while there are plenty of things I could use over there, none quite solve my problem. The primary interest here is with structure. I basically have a complex (representing relationships) and a function $f \, : \, X \to \mathbb{R}$ that assigns a value for each open $U \in X$. (This is why I thought sheaf theory might be helpful.) But to be honest I'm quite lost. Just the rank of $\pi_1$ could be enough, but I thought I could get further with this! Mar 17 comment How can I understand cohomology theories in the context of basic homology theory? I think I might need to re-post the question framed in a different way. I might be looking at the wrong mathematical tool. I thought perhaps sheaf cohomology would have something to do with it. Basically, the goal is to capture the connectivity structure of a graph, but to use the weights on its edges to derive some kind of "metric." The reason we care about things like "loops" and such is kind of complex, and has to do with a behavioral model, I'm really just searching for the right way to handle a problem in which one wants to quantify the above structure in some rigorous (algebraic) way. Mar 17 comment How can I understand cohomology theories in the context of basic homology theory? Thank you for the clarifications. Although I'm still not clear on how it is that cohomology attaches a notion of "quantity" while homology does not. For example, Wikipedia says "cochains in the fundamental sense should assign 'quantities' to the chains of homology theory" -- what do you make of that? Mar 17 awarded Curious Mar 16 asked How can I understand cohomology theories in the context of basic homology theory? Jan 6 awarded Yearling Jan 6 comment show H is closed under * @JorgeFernández totally dig that but thought it might help if he had something to Google... and I don't think the argument relies on anything group-theoretic, just the associativity and commutativity of $*$ Jan 6 answered show H is closed under * 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 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. :)