## Galois Theory Simplified Math Online Tom Circle

Galois discovered Quintic Equation has no radical (expressed with +,-,*,/, nth 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{ikpi}{5}}\$
\$latex text{where k } in {0,1,2,3,4}\$
which can be expressed in
\$latex x= cos frac{kpi}{5} + i.sin frac{kpi}{5} \$
hence in {+,-,*,/, √ }
ie
\$latex x_0 = e^{frac{i.0pi}{5}}=1\$
\$latex x_1 = e^{frac{ipi}{5}}\$
\$latex x_2 = e^{frac{2ipi}{5}}\$
\$latex x_3 = e^{frac{3ipi}{5}}\$
\$latex x_4 = e^{frac{4ipi}{5}}\$
\$latex x_5 = e^{frac{5ipi}{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 =…

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