Neocities.org
>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
3 weeks 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
3 weeks 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)
10,000 views! pretty cool
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eastathenaeum
3 weeks ago
I love how you've presented the circular and hyperbolic page, despite being told f(u). Lol
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xpaper
1 month ago
The wall of text I said I would add is now here! It is still incomplete however. I admit the picture I chose for this article might be a bit provocative lol.
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xpaper
1 month ago
In true mathematics fashion, I decided to leave the details I was too lazy to write as an exercise to the reader. Now only the Spivak citation is missing... Anyways, back to my unrelated wall of text. ken la jan pi toki pona li olin ala e ona!
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xpaper
1 month ago
Also, removed link to Differential Forms since I plan on doing something different with regards to that topic now.
Merry Christmas! I have decided to delay completing the Trig Notes since the only thing missing is a proof that sinh/cosh are odd/even functions and I want to get to my other plans, not to mention that I hope to overhaul the notes after learning complex analysis. Stay tuned for a wall of text on an unrelated topic!
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The main problem with doing this right now is that I don't know Galois theory :)