Techniques for Solving Solvable Equations via Galois Theory

Table of Contents
[Ch.1]	Mastering Galois-style Equation Solving at Lightning Speed	\(C_2\)	\(x^2+x+1\)	2pp.
[Ch.2]	The Essence of Galois Theory Packed Into One	\( S_3 \)	\(x^3+3x+1 \)	9pp.
[Ch.3]	Exhausting! Packed with Cyclic Extensions!	\( S_4 \)	\( x^4+4x+2 \)	12pp.
[Ch.4]	Super Cool! The Magic Technique of Resultants	\(A_3\)	\(x^3-3x+1\)	4pp.
[Ch.5]	The Terror of Factorization: Trager Algorithm	\( A_3\)	\(x^3-3x+1 \quad (gcd)\)	5pp.
[Ch.6]	Solving Cyclic Equations \(\Phi_{5}(x)\)	\( C_4 \)	\(x^4+x^3+x^2+1 \)	3pp.
[Ch.7]	Galois-Style Solution for the Cyclotomic Equation \(\Phi_{17}(x)\)	\( C_{16} \)	\(x^{16}+x^{15}+...+x^2+x+1 \)	6pp.
[Ch.8]	A Treasure Trove of Group Theory Problems: Frobenius Groups	\( F_{20} \)	\( x^5+x^4+2x^3+4x^2+x+1\)	10pp.
[Ch.9]	What!? Is the Galois-Style Method Falling Apart?	?,?	\(x^3-2, x^5-5x^3+5x+6 \)	5pp.

(Ultra-fast track: Chapter 1), (Fast track: Chapters 2, 4, and 5), (Resultant course: Chapters 4, 5, and 9)

◆ Regarding navigation on this site:

(1) Clicking on [Ch. **] in the table of contents above will take you to the corresponding chapter.

(2) Clicking the arrows ⇦ Home ⇨ at the top-right of each page, and just below the explanation on the left, will take you to the previous page, the home page, or the next page, respectively.

(3) The two ▶ symbols in the light-green box at the top of each page have the following roles:
• ▶ Page 1,2,3,..,n — clicking a number will take you to that page.
• ▶ Sample Program — clicking this will open, in a new tab, a PDF of the Maxima program for the solution steps in the corresponding chapter. Please note, however, that these are extremely simple programs created by an amateur, so I ask for your understanding.

◆ Purpose of this site

Have you perhaps grown weary of Galois theory explanations that go "theorem, proof, theorem, proof" in endless succession? The unique feature of this site is that there are absolutely no theorems or proofs. Instead, we focus entirely on introducing Galois's equation-solving techniques in detail, using seven examples.

To master these computational techniques, it is important to perform the calculations yourself using algebraic computation software. Even solving just one problem is fine — try working through it while following the explanations.

For that purpose, we’ve tried to present all calculation steps as fully as possible, so you can directly compare your results with those shown on the site. Doing this will gradually allow you to *feel* the meaning behind Galois theory's theorems.
Wouldn’t it be fun to master Galois-style equation-solving techniques and be able to solve equations with ease?

◆ Excerpt from This Website

These Are the Only Techniques You Need
for Solving Equations! (Chapter 1)

\begin{align} &f(x)=x^2+x+1 \quad v \equiv \alpha+2\beta \notag \\ \notag \\ &V(x) \equiv (x-v_{1})(x-v_{2})=g_0(x) \notag \\ &P_\alpha(x)=V(x) \bigl(\frac{\alpha }{x-{v_1}}+\frac{\beta }{x-{v_2}} \bigr) \notag \\ &\alpha=\frac{ P_\alpha(v_1)}{(v_1-v_2)} \notag \\ \notag \\ &\left\{ \begin{array}{l} h_0 = (x-v_1) \\ h_1 = (x-v_2) \\ \end{array} \right. \notag \\ &\begin{bmatrix} t_0 \\ t_1 \end{bmatrix} \equiv \frac{1}{2} \begin{bmatrix} 1&1 \\ 1&-1 \end{bmatrix} \cdot \begin{bmatrix} h_0 \\ h_1 \end{bmatrix} \notag \\ \notag \\ &\left\{ \begin{array}{l} B_0(x)=x^2-A_0=0 \\ t_1^2=A_0=-\frac{3}{4} \\ a_0=\sqrt{A_0} \ \in Q_1 \\ \end{array} \right. \notag \\ \notag \\ &\begin{bmatrix} \tilde{h_0} \\ \tilde{h_1 } \end{bmatrix} = \begin{bmatrix} 1&1 \\ 1&-1 \end{bmatrix} \cdot \begin{bmatrix} t_0 \\ \tilde{t_1} \end{bmatrix} = \begin{bmatrix} g_1(x) \\ t_0- \tilde{t_1} \end{bmatrix} \notag \\ \notag \\ &\left\{ \begin{array}{l} g_1(x)=0 \ \rightarrow \ v=-\frac{3}{2}-a_0 \\ \therefore \alpha=-v-2=-\frac{1}{2}+a_0 =\omega \end{array} \right. \notag \\ \end{align}

\(\quad\)

Trager Algrithm Color-Coded for Clarity,
Since It’s Complicated (Chapter 5)

\begin{align} &f(x)=x^3-3x+1 \quad g(v)=v^3-9v-9 \notag \\ \notag \\ &Res(f(x+v) ,g(v),v) \notag \\ &=-(x^3-21x+37)(x^3-12x-8)(x^3-3x+1) \notag \\ \notag \\ \end{align}

\begin{align} Res&(f(x+v), g(v),v) \notag \\ =(-1)& \left\{\bbox[#FFFF00]{ (x+v_1-\alpha) }\bbox[#7FFF00]{(x+v_1-\beta)}\bbox[#00FFFF]{(x+v_1-\gamma)} \right \} \notag \\ \times &\left\{\bbox[#00FFFF]{ (x+v_4-\alpha)}\bbox[#FFFF00]{ (x+v_4-\beta)}\bbox[#7FFF00]{(x+v_4-\gamma)} \right \} \notag \\ \times &\left\{\bbox[#7FFF00]{(x+v_5-\alpha)}\bbox[#00FFFF]{(x+v_5-\beta)}\bbox[#FFFF00]{ (x+v_5-\gamma)} \right \} \notag \\ \notag \\ =(-1)& \left\{ \bbox[#FFFF00]{(x+v_1-\alpha)(x+v_4-\beta)(x+v_5-\gamma)} \right \} \notag \\ \times &\left\{\bbox[#7FFF00]{(x+v_1-\beta)(x+v_4-\gamma)(x+v_5-\alpha)} \right \} \notag \\ \times &\left\{\bbox[#00FFFF]{(x+v_1-\gamma)(x+v_4-\alpha)(x+v_5-\beta)} \right \} \notag \\ \notag \\ =(-1)& \cdot\bbox[#FFFF00]{H_1} \cdot \bbox[#7FFF00]{H_2} \cdot \bbox[#00FFFF]{H_3} \notag \\ \notag \\ \end{align}

\begin{align} &f(x+v) \notag \\ &=\bbox[#FFFF00]{(x+v-\alpha)} \cdot \bbox[#7FFF00]{(x+v-\beta)} \cdot \bbox[#00FFFF]{(x+v-\gamma)} \notag \\ \notag \\ &\left\{ \begin{array}{l} GCD\bigl(\left. \bbox[#FFFF00]{H_1} \right|_{v_1=v} \ ,f(x+v)\bigr)= \bbox[#FFFF00]{(x+v-\alpha)}=Y_1 \\ GCD\bigl(\left. \bbox[#7FFF00]{H_2} \right|_{v_1=v} \ ,f(x+v)\bigr)= \bbox[#7FFF00]{(x+v-\beta)}=Y_2 \\ GCD\bigl(\left. \bbox[#00FFFF]{H_3} \right|_{v_1=v} \ ,f(x+v)\bigr)= \bbox[#00FFFF]{(x+v+\gamma)}=Y_3 \\ \end{array} \right. \notag \\ \notag \\ \end{align}

\begin{align} f(x)&=Y_1(x-v) \cdot Y_2(x-v) \cdot Y_3(x-v) \notag \\ &=(x-x_1)(x-x_2)(x-x_3) \notag \\ \end{align}

Cyclic Extensions and Composition Series (Chapter 8)

◆ Sites I Used as References

・lemniscus – Recursive Iteration Blog “From Equations to Galois Theory” https://lemniscus.hatenablog.com/entry/20120527/1338129004
・Keita Ikumi – “How to Find the Galois Group of an Equation: Adventures in a Fifth-Dimensional World” https://ikumi.que.jp/blog/archives/252
・"Amateur Mathematician After Retirement" – “Solving Solvable Algebraic Equations Based on Galois Theory” https://ikumi.que.jp/blog/wp-content/uploads/2019/09/galois-solution-ver2.pdf
・jurupapa – “Math with Maxima” https://maxima.hatenablog.jp/entry/2017/10/21/113926

The authors of these four sites are benefactors who guided me into a world I had never known. By reading their explanations and actually doing the calculations myself, the many theorems and terms of Galois theory finally connected for the first time.