This document discusses solving quintic (degree 5) polynomial equations. It presents the general form of a quintic polynomial and methods for finding its roots, including using the factorization of polynomials and representations of the roots in terms of radicals. It also discusses representing functions as tensors and decomposing them into products of simpler tensors.

1. tsujimotter
12 # 2018/06/16
x5
x + a = 0

2. x5
x + a = 0

3. x2
+ ax + b = 0
x =
a ±
p
a2 4b
2

4. x =
a ±
p
a2 4b
2
x2
1 = a2
4b
x2 =
a
2
+
x1
2

5. xk2
2 = f2(a, x1)
xk1
1 = f1(a)
xkN
N = fN (a, x1, . . . , xN 1)
...
x5
N xN + a = 0
x5
x + a = 0

6. a = 0 x5
x = (x 1)(x + 1)x(x i)(x + i) = 0
i
i
1 10
x5
x + a = 0

7. x5
x + a = 0a

8. x5
x + a = 0a

9. x5
x + a = 0a

10. x5
x + a = 0a

11. x5
x + a = 0a

12. 1
(i) 1 2
2

13. 2
3
1
(ii) 1, 2, 3

14. http://tsujimotter.info/works/quintic/

15. •
n 5
• N
• N xN
• xN
•

16. 1 2 3 4 5
1 2 3 4 5
(2 3 1 4 5)

17. 1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
= (2 3 1 4 5)
= (1 4 3 2 5)
= (4 3 1 2 5)

18. [↵, ] := ↵ ↵ 1 1

19. n 5
↵0 = [↵1, ↵2][↵3, ↵4] · · · [↵2s 1, ↵2s]

20. ↵0 = [↵1, ↵2][↵3, ↵4] · · · [↵2s 1, ↵2s]

21. ↵0 = [↵1, ↵2][↵3, ↵4] · · · [↵2s 1, ↵2s]
↵1 = [↵1,1, ↵1,2][↵1,3, ↵1,4] · · · [↵1,2t 1, ↵1,2t]

22. ↵0 = [↵1, ↵2][↵3, ↵4] · · · [↵2s 1, ↵2s]
↵1 = [↵1,1, ↵1,2][↵1,3, ↵1,4] · · · [↵1,2t 1, ↵1,2t]
↵1,1 = [↵1,1,1, ↵1,1,2] · · · [↵1,1,2r 1, ↵1,1,2r]

23. ↵0 = [↵1, ↵2][↵3, ↵4] · · · [↵2s 1, ↵2s]
↵1 = [↵1,1, ↵1,2][↵1,3, ↵1,4] · · · [↵1,2t 1, ↵1,2t]
...
↵1,...,1 = [↵1,1,...,1, ↵1,1,...,2] · · · [↵1,1,...,2z 1, ↵1,1,...,2z]
↵1,1 = [↵1,1,1, ↵1,1,2] · · · [↵1,1,2r 1, ↵1,1,2r]

24. ↵0 = [↵1, ↵2][↵3, ↵4] · · · [↵2s 1, ↵2s]
↵1 = [↵1,1, ↵1,2][↵1,3, ↵1,4] · · · [↵1,2t 1, ↵1,2t]
...
↵1,...,1 = [↵1,1,...,1, ↵1,1,...,2] · · · [↵1,1,...,2z 1, ↵1,1,...,2z]
↵1,1 = [↵1,1,1, ↵1,1,2] · · · [↵1,1,2r 1, ↵1,1,2r]

25. 0
`0 = [`1, `2][`3, `4] · · · [`2s 1, `2s]
`1 = [`1,1, `1,2][`1,3, `1,4] · · · [`1,2t 1, `1,2t]
`1,1 = [`1,1,1, `1,1,2] · · · [`1,1,2r 1, `1,1,2r]
`1,...,1 = [`1,1,...,1, `1,1,...,2] · · · [`1,1,...,2z 1, `1,1,...,2z]
...

26. xk2
2 = f2(a, x1)
xk1
1 = f1(a)
xkN
N = fN (a, x1, . . . , xN 1)
...
x5
N xN + a = 0

27. xk2
2 = f2(a, x1)
xk1
1 = f1(a)
xkN
N = fN (a, x1, . . . , xN 1)
...
x5
N xN + a = 0
a

28. xk1
1 = f1(a) x1, x1⇣, x1⇣2
, · · · , x1⇣k1 1

29. xk1
1 = f1(a) x1, x1⇣, x1⇣2
, · · · , x1⇣k1 1
`1 x1 7! x1⇣m1
x1 7! x1⇣m2
`2
[`1, `2] = `1`2` 1
1 ` 1
2

30. xk1
1 = f1(a) x1, x1⇣, x1⇣2
, · · · , x1⇣k1 1
x1
`1 x1 7! x1⇣m1
x1 7! x1⇣m2
`2
[`1, `2] = `1`2` 1
1 ` 1
2
x1
`1
7 ! x1⇣m1
`2
7 ! x1⇣m1+m2
` 1
1
7 ! x1⇣m2
` 1
2
7 ! x1

31. x1

32. x1
x2

33. x1
N-1
xN-1
N
xN

34. N
xN
xN

35. N
xN
xN

36. •
n 5
• N
• N xN
• xN
•

37. D. S.
(2012)