xpaper
11 months ago
Given that I haven't updated the notes on the circular functions in ages, I decided I might as well start making the translations. First, Spanish
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xpaper
1 year ago
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.
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xpaper
1 year ago
I AM ALIVE and I have once again decided to overhaul the navigation because I am indecisive.
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xpaper
1 year 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
1 year 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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![[tfpXE] avatar](/site_screenshots/15/25/tfpxe/index.html.50x50.webp)
I've just started learning Esperanto!
Very excited to hear that!