**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):

**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]

**Cubic Equation**: 16 CE Italians del Ferro, Tartaglia & Cardano

Cardano Formula (1545 《Ars Magna》):

**Example**:

By obvious guess, x = 4

Using Cardano formula,

They discovered the first time in history the *“Imaginary” number* (aka **Complex** number):

then

Quartic Equation: by Cardano’s student Ferrari

**Quintic Equation**:

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 , Real , Complex , (Integer modulo prime, eg.Z2 = {0, 1}) , etc.

If (“a”, “b”) is adjoined with irrational (eg. ) to become a **larger Field (extension) **

it has a beautiful “**Symmetry**” aka **Conjugate**

Field Extension of :

Any equation P(x) = 0

with root in will have

another **conjugate** root

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]