5차방정식과 근의 공식

 

 

 

 

증명의 개요

• We start from the field of symmetric functions.
  
• Essentially, we are studying the radical extension of that base field.
  
• The proof is consisted of two steps.
  
• radicals to express the quintic formula can be expressed in terms of roots
  
• the behavior of radicals under permutations
  

 

 

 

오차방정식

 

 

• \(x^5 - s_{1} x^{4} + s_{2} x^{3} -s_{3}x^{2}+s_{4} x - s_5= 0\)
  

 

 

• \(K=\mathbb{C}(x_1,x_2\cdots,x_5)\)
  
• \(F=\mathbb{C}(s_1,s_2,\cdots,s_n)\)
  

 

 

• 갈루아 이론
  
• 갈루아 이론 (피)
  

 

 

 

 

 

Monodromy proof

Consider \(3w^5-25w^3+60w-z=0\).

For \(z=\pm 38\) and \(z=\pm 16\), the above equation has four distinct roots.

These are the branch points and determines the Riemann surfaces.

Then the monodromy group is acting as a permutation of sheets and not solvable.

(This is a little different from the Galois group.)

We can apply this monodromy idea to the computation of Galois groups of number fields.

 

 

regular proof

\(f(x)=2x^5-5x^4+5\) is the irreducible polynomial of degree 5 over the rationals.

It has two complex and 3 real roots.

This implies the Galois group is \(S_5\).

 

 

일반적인 n차 방정식

 

일반적인 방정식

\(x^n - s_{1} x^{n-1} + s_{2} x^{n-2} + \cdots + (-1)^{n-1}s_{n-1} x +(-1)^n s_n= 0\)

 

\(K=\mathbb{C}(x_1,\cdots,x_n)\)

\(F=\mathbb{C}(s_1,\cdots,s_n)\)

 

 

관련된 학부 과목과 미리 알고 있으면 좋은 것들

• 추상대수학
• 오차방정식과 정이십면체

 

 

 

관련된 다른 주제들

• 추상대수학의 토픽들
•  

 

 

링크

• http://en.wikipedia.org/wiki/Abel–Ruffini_theorem
• http://fermatslasttheorem.blogspot.com/2008/10/abels-impossibility-proof.html
  

 

 

관련논문

• Abel's Proof
  
  • Peter Pesic, Chapter 6. 'Abel's proof' 85-94p (pdf)
• Galois' Theory of Algebraic Equations
  
  • Jean-Pierre Tignol, Chapter 13.  Ruffini and Abel on general equations (pdf)
• Elliptic functions and elliptic integrals[1]
  
  • Viktor Prasolov, Yuri Solovyev, 6.5 The Abel theorem on the solvability in radicals of the general quinti equation (pdf)
• Variations on the theme of solvability by radicals
  
  • A. G. Khovanskii, Proceedings of the Steklov Institute of Mathematics,     Volume 259, Number 2 / 2007년 12월
• On solvability and unsolvability of equations in explicit form
  
  • A G Khovanskii, Russian Math. Surveys 2004, 59 (4), 661-736
• Niels Hendrik Abel and Equations of the Fifth Degree
  
  • Michael I. Rosen, The American Mathematical Monthly, Vol. 102, No. 6 (Jun. - Jul., 1995), pp. 495-505