In Search for Radical Roots of Polynomial Equations of degree n > 1

Take note: Find roots (根) to solve polynomial (多项式方程式) equations, but find solutions (解) to solve simultaneous equations (联式方程式).

Radical : (Latin Radix = root): \sqrt [n]{x}

Quadratic equation (二次方程式) [最早发现者 : Babylon  和 三国时期的吴国 数学家 赵爽]

{a.x^{2} + b.x + c = 0}

\boxed{x= \frac{-b \pm \sqrt{b^{2}-4ac} }{2a}}

Cubic Equation: 16 CE Italians del Ferro,  Tartaglia & Cardano
{a.x^{3} = p.x + q }

Cardano Formula (1545 《Ars Magna》):
\boxed {x = \sqrt [3]{\frac {q}{2} + \sqrt{{ (\frac {q}{2})}^{2} - { (\frac {p}{3})}^{3}}} + \sqrt [3]{\frac {q}{2} -\sqrt{ { (\frac {q}{2})}^{2} - { (\frac {p}{3})}^{3}}}}

Example:
{x^{3} = 15x + 4}
By obvious guess,  x = 4
Using Cardano formula,
x = \sqrt[3]{2+ 11 \sqrt{-1}} + \sqrt [3]{2 - 11 \sqrt{-1}}

They discovered the first time in history the “Imaginary” number (aka Complex number):
\boxed {i = \sqrt{-1}}
then
(2 + i)^{3} =2+11i
(2 - i)^{3} =2-11i
x = (2 + i) + (2 - i) = 4

Quartic Equation: by Cardano’s student Ferrari
{a.x^{4} + b.x^{3} + c.x^{2} + d.x + e = 0}

Quintic Equation:
{a.x^{5} + b.x^{4} + c.x^{3} + d.x^{2} + e.x + f = 0}

No radical solution (Unsolvability) was suspected by Ruffini (1799), proved by Norwegian Abel (1826), but explained by French 19-year-old boy Évariste Galois (discovered in 1831, published only after his death in 1846) with his new invention : Abstract Algebra “Group“(群) & “Field” (域)。

Notes:

Group Theory is Advanced Math.
Field Theory is Elementary Math.

Field is the Algebraic structure which has 4 operations on calculator (+ – × ÷). Examples : Rational number (\mathbb{Q}), Real (\mathbb{R}), Complex (\mathbb{C}), \mathbb{Z}_{p}  (Integer modulo prime, eg.Z2 = {0, 1}) , etc.

If \mathbb{Q}   (“a”, “b”) is adjoined with irrational (eg. \sqrt {2})  to become a larger Field (extension) \mathbb{Q} (\sqrt {2}) = a +b\sqrt {2}
it has a beautiful “Symmetry” aka Conjugate
(a - b\sqrt {2}) 

Field Extension of \mathbb{Q} (\sqrt {2}) = a +b\sqrt {2} :

Any equation P(x) = 0
with root in \mathbb{Q} (\sqrt {2}) = a +b\sqrt {2} will have
another conjugate root (a - b\sqrt {2})

Galois exploited such root symmetry in his Group structure to explain the unsolvability for polynomial equations of quintic degree and above.

Ref: 《Elements of Mathematics – From Euclid to Gödel》by John Stillwell (Princeton University Press, 2016) [NLB # 510.711]

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