mebassett
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 Jan26 comment Category of $\textbf{Ring}$. it's sometimes easier to have a convention that excludes slightly trivial or slightly pathological examples. Dummit and Foote just take a ring to mean something with a unit, they never deal with unit-less rings, and it's extra baggage to carry that around when you aren't using it. Jan26 comment Univariate and Matrix Representation of Affine Transformation the map $\phi$ is implicit in what I'm doing, it's essentially a change-of-basis matrix. if you let $e_i$ be the canonical basis (a column with all 0s except on the i-th row, that's a 1) for $\mathbb{F}^n$, then $\phi : \mathbb{F}^n \to \mathbb{E}^n$ by $e_i \mapsto \{\text{some expression in the t basis I gave}\}$ then it'll do the trick. Jan25 comment Univariate and Matrix Representation of Affine Transformation I'm not 100% sure I'm correct on the last step, but hopefully you see the idea now? :) Jan25 comment Application of Category Theory you might look at these questions: math.stackexchange.com/questions/312605/… and mathoverflow.net/questions/19325/… (heavier maths there) Jan14 comment most efficient way to find distinct complimenting subspaces over a finite field is there an algorithmically efficient way to complete $\{v_{n-k+1},\ldots, v_n\}$ to a basis $\{v_1,\ldots,v_n\}$. Your description solves the problem wonderfully (though I no longer need such a solution!), but I might be too silly to see an way to complete the basis. Jan14 comment most efficient way to find distinct complimenting subspaces over a finite field you're correct, my bad. changed question. Dec31 comment Is there a discrete version of non-commutative geometry (yet)? you might be interested in a paper from Majid arxiv.org/abs/1011.5898 Noncommutative riemannian geometry on graphs. This is not NCG in the manner of connes, though. Dec9 comment finding fixed points of frobenius endomorphism of a ring of char > $p$ (not a domain) this is a very good answer. but it leaves me with a lot of nagging questions: why does the number of cycles = the number of distinct irreducible factors of $x^{p^n} -x$? Why does the number of cycles of length $d$ = the number of such factors of degree $d$? I suppose these things deserve separate questions and I should accept your answer - is that the proper etiquette here? Jan5 comment Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$. certainly! google my name on here and you should find my contact info. Dec28 comment Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$. I'm working through majid's "primer on quantum groups" (disclaimer: majid is my phd supervisor) along with kassel, and I have a few other texts handy, but they're harder for an undegrad to crack. feel free to email me about it, though. Dec17 comment Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$. a word of warning about kassel - it's a great book, but he tends to be a bit "over axiomatic" and it's a bit hard to see the forest through the trees sometimes. For instance, it's a lot easier to just compute $\Delta(\text{det_q})$ then to follow his chain of propositions. It helps to have a few other texts handy to cross-reference these things. Dec17 comment Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$. I re-wrote it a bit more coherently, also describing coideals a tad bit more and explaining how he gets the computation. You should ensure you have a good idea of ideals and quotient rings if you don't have that already. coideals were described in ch3. I'm a 2nd year phd student and I find this hopelessly confusing myself. regards! Dec17 comment Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$. do you understand how the bialgebra structure on $M_q(2)$ works? Dec17 comment proving that a action of hopf algebra k(G) on A implies a G-grading on A I added some text for clarification, does that help? Aug6 comment What are some examples of vector spaces that aren't graded? well that certainly clears things up. thanks! Jul5 comment what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$? this is indeed a helpful answer, and sheds much light on the proof in conway's book I didn't understand (he uses the same technique with that polynomial). In fact it solves my problem for all finite numbers where its possible to solve it. But I'd still like a description of the other subfields, though. I'm going to leave the question open, but if you're interested I'll post a note or another question on the bits I [do not] understand from Conway's book. Thanks! :D Jul5 comment About PhD in non-commutative topology You may want to try reading something like Kalkhali's Very Basic Noncommutative Geometry arxiv.org/abs/math/0408416. Though in my (very limited) experience it's more effective to approach research by trying to tackle a specific problem and learning the tools you need as you go along. In lieu of that, read as much operator algebra and geometry as you can. Jul5 comment what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$? Oops. Never mind, such fields are covered by infinite ordinals. This is why I should wait to respond! Jul5 comment what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$? One thing is bothering me, though, and this may be a separate question: if $\omega^{\omega^\omega}$ is an algebraic closure, and if an algebraic closure contains all $\mathbb{F}_{2^k}$, even k not a power of 2, then shouldn't $\omega^{\omega^\omega}$ contain a subfield isomorphic to $\mathbb{F}_{2^k}$? Indeed this is hard to reconcile with the multiplication table at me neverendingbooks for say, k=3. Ordinals $2^{2^k}$ only give us a quadratic closure, no? Jul5 comment what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$? This is nevertheless a helpful answer! I'll comment more ( or accept) later today when I can read over it a bit more carefully.