5차방정식과 근의 공식

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.

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

It has two complex and 3 real roots.

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

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.

1. radicals to express the quintic formula can be expressed in terms of roots

2. the behavior of radicals under permutations

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

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

step 1

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[2]
- 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

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