Galois Theory Simplified

Galois discovered Quintic Equation has no radical (expressed with +,-,*,/, square root) solutions, but his new Math "Group Theory" also explains:
$latex x^{5} - 1 = 0 \text { has radical solution}$
but
$latex x^{5} -x -1 = 0 \text{ has no radical solution}$

Why ?

$latex x^{5} - 1 = 0 \text { has 5 solutions: } \displaystyle x = e^{\frac{ik\pi}{5}}$
$latex \text{where k } \in \{0,1,2,3,4\}$
which can be expressed in
$latex x= cos \frac{k\pi}{5} + i.sin \frac{k\pi}{5} $
hence in {+,-,*,/, √ }
ie
$latex x_0 = e^{\frac{i.0\pi}{5}}=1$
$latex x_1 = e^{\frac{i\pi}{5}}$
$latex x_2 = e^{\frac{2i\pi}{5}}$
$latex x_3 = e^{\frac{3i\pi}{5}}$
$latex x_4 = e^{\frac{4i\pi}{5}}$
$latex x_5 = e^{\frac{5i\pi}{5}}=1=x_0$

=>
$latex \text {Permutation of solutions }{x_j} \text { forms a Cyclic Group: }
\{x_0,x_1,x_2,x_3,x_4\} $

Theorem: All Cyclic Groups are Solvable
=>
$latex x^{5} -1 = 0 \text { has radical solutions.}$

However,
$latex x^{5} -x -1 = 0 \text{ has no radical solution } $
because the permutation of solutions is A5 (Alternating Group) which is Simple
ie no Normal Subgroup
=> no Symmetry!

Coset

The powerful notion of Coset was invented by Galois (l'ensemble à gauche ou à droit), but only named as Coset after 150+ yrs later by G.A. Miller in 1910.

Prove:
Coset * Coset = Coset
=> Normal Subgroup

[Hint] Proof technique: use
1) $latex a^{-1}$
2) e

Proof:
1) For any a ∈G, H subgroup of G,

$latex (Ha)(Ha^{-1})= H.(aHa^{-1})$

2) Given $latex H.(aHa^{-1}) $ is right coset,
Choose $latex (aHa^{-1}) = e \in G$
$Latex H.(aHa^{-1})= He = H$
=> $latex aHa^{-1} \subset H$
=> H Normal subgroup

Galois's 1st Public Lecture

[caption id="" align="alignright" width="300"] Évariste Galois (1811–1832) (Photo credit: Wikipedia)[/caption]

19-yr-old Galois's 1st public Math Tuition class after being expelled from Ecole Normale Superieure:

"My work proceeds from a tendency to think intuitively. Dispensing the details, I prefer to leap from foresight to idea to concept...
I propose an analysis of analysis : a new, terser (abstract) algebra, where the most elevated calculations will be considered as particular case, which ultimately must be put aside for greater, more general investigations... so my foresight will be used by generations of mathematicians to advance new theories."