純粋・応用数学・数学隣接分野（含むガロア理論）12

[過去ﾛｸﾞ] 純粋・応用数学・数学隣接分野（含むガロア理論）12 (1002ﾚｽ)
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506: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2023/01/08(日) 13:50:02.24 ID:9zXu/9tz

>>503 補足

月うさぎレベルよりましな話なら、下記があるよ
しかし、それ以上の具体的話は、ない
1の11乗根のべき根表示に、具体的に役立つ話は、ない

そもそも、”Roots of unity are used in many branches of mathematics”,”the theory of group characters, and the discrete Fourier transform.”
それ以上のなにか、ある？ｗ

https://en.wikipedia.org/wiki/Root_of_unity
Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

https://en.wikipedia.org/wiki/Character_theory
Character theory
This article is about the use of the term character theory in mathematics. For related senses of the word character, see Character (mathematics).
　”https://en.wikipedia.org/wiki/Character_(mathematics)
　In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings.[1] Other uses of the word "character" are almost always qualified.
　https://ja.wikipedia.org/wiki/%E6%8C%87%E6%A8%99_(%E6%95%B0%E5%AD%A6)
　指標 (数学)
　少なくとも二つの、異なるが重複もする意味が存在する。
　乗法的指標
　表現の指標”

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. T
Contents
4 Character tables
4.2 Character table properties
Character table properties
This gives rise to a group of linear characters, called the character group under the operation [Χ1*Χ2](g)=Χ1(g)Χ2(g).
This group is connected to Dirichlet characters and Fourier analysis.

http://rio2016.5ch.net/test/read.cgi/math/1671460269/506

626: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2023/01/09(月) 20:12:42.24 ID:xY+wMPX4

>>595 追加
>https://joelshapiro.org/Pubvit/Downloads/Rulla_dftradicals.pdf
>Solution by Radicals and the DFT January 11, 2019 Jim Rulla

（DFTとLagrange resolventとの関係）
これのP8より
The DFT of the roots is
＜下記は行列です。原文ご参照！＞
[1  1　1 1](r1)＝(r1 + r2 + r3 + r4) ・・・・・(B4)
[1 -i -1 i](r2)　(r1 - r2i - r3 + r4i)
[1 -1 1-1](r3)　(r1 - r2 + r3 - r4)
[1 i -1 -i](r4)　(r1 + r2i - r3 - r4i)
The top row, as always, is symmetric in the roots, and is - b.
The second and fourth rows are similar ? they both qualify as Lagrange resolvents ? but the third row is different.
(引用終り)

（コメント）
・ここで、トップの1行目は、単純な根の和で、-bです
・2行目と4行目は、Lagrange resolvent
・3行目は、Lagrange resolventではない！
・つまり、上記のDFTの行列は、Lagrange resolvent そのものではない！！
（Lagrange resolventの拡張と言えるかもね）

なお、Lagrange resolvent で、4次方程式が解けることは
P9 の冒頭 Remark で
”One can also solve the quartic using the Lagrange resolvents.See Edwards6.”
と記されている。（Edwards 6でなくても、他にもありと思うけど）
以上

http://rio2016.5ch.net/test/read.cgi/math/1671460269/626

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