Neocities.org
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xpaper
2 weeks ago
I AM ALIVE and I have once again decided to overhaul the navigation because I am indecisive.
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xpaper
2 months ago
redid the UI again so it takes up less space. if you're seeing that the site content takes up almost no space, the thumbnail just needs to refresh
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Finally overhauled the website. Now everything is in an iframe and there's a bar at the top for navigation.
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IM ALIVE.
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xpaper
3 months ago
I'll get back to updating the site eventually, but i think the whole thing could use an overhaul
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>the_quintic_problem By the way historically the origin was the quintic problem. But that era has ended and Galois theory is used in number theory the most. I could even say that it is basically linear algebra for number theory. So study hard if you were interested in number theory.
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>the_quintic_problem Therefore an equation of degree n has general solutions if and only if Sn is solvable. And since Sn is a finite group, it means whether Sn has a composition series in which all factor groups are Zp. This corresponds to that by adding the primitive p-th root of unity, you can extend the base field to the field that factorizes the polynomial. But for n>4, An is simple so that Sn is not solvable.
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>the_quintic_problem I will not explain long. The solutions for a cubic equation depend on a+bw+gw^2 (Lagrange resolvent) and the permutations of {1,w,w^2}. Because in the end those permutations form the Galois group which fixes the base field. So it all depends on the structure of S3. Similarily for a quartic equation, {1,i,i^2,i^3} and its permutations S4. And so on.
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For the (potentially distant) future of the Math Notes, I hope to write notes on abs. algebra. It is disappointing that despite the quintic problem being one of the origins of abs. alg., 1st semester courses rarely talk much about it. I'm about to start a course on Galois theory. I hope to afterwards write notes which use this problem as a motivation for the concepts of group theory, etc.
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xpaper
5 months ago
The main problem with doing this right now is that I don't know Galois theory :)
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xpaper
5 months ago
I said I had big plans for this website after the semester ended, and then didn't do any of them. Oops! Well, anyways, here are some of them now that the next semester is about to start.
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xpaper
5 months ago
Changelog: Added embeds for more music, started a project to centralize and translate original Toki Pona works (so long as their license allows it, of course)
decided to add a funny thing on one of the pages. whatever. one day i'll make an actually good update on the math notes or something one day i promise. youtube has me distracted.