The 9th Roots of Unity¶

The ninth-root equation factors as

$$ x^9-1=(x^3-1)(x^6+x^3+1). $$

We solve the cyclotomic factor $\Phi_9(x)=x^6+x^3+1$. For a primitive root $\zeta=\exp(2\pi i/9)$, the result implies

$$ \cos\frac{2\pi}{9} =\frac{\sqrt[3]{\omega}+\sqrt[3]{\omega^2}}{2}, \qquad \omega^3=1,\ \omega\ne1. $$

Primitive conjugate pairs¶

The primitive exponent classes modulo $9$ are $1,2,4,5,7,8$. Pairing inverses produces three real periods. Their sum is zero, in agreement with the vanishing sum of primitive ninth roots.

def z : MathValue := rtu 9

def a11 : MathValue := z ^ 1 + z ^ 8
def a12 : MathValue := z ^ 2 + z ^ 7
def a13 : MathValue := z ^ 4 + z ^ 5

def b10 : MathValue := a11 + a12 + a13

def cyclotomic9 : MathValue :=
  '((rtu 9)^6 + (rtu 9)^3 + 1)

def reduce9 (v : MathValue) : MathValue :=
  idealNFWith [w] [cyclotomic9] v

def b10' : MathValue := reduce9 b10

b10'

$0$

A modern three-point resolvent¶

Multiplication of exponents by $2$ cycles the three periods. The following typed definitions are the two nontrivial Fourier resolvents for that cycle. This is a fresh translation into the current Egison syntax.

def b11 : MathValue := a11 + w * a12 + w ^ 2 * a13
def b12 : MathValue := a13 + w * a11 + w ^ 2 * a12
def b13 : MathValue := a12 + w * a13 + w ^ 2 * a11

def b14 : MathValue := a11 + w * a13 + w ^ 2 * a12
def b15 : MathValue := a12 + w * a11 + w ^ 2 * a13
def b16 : MathValue := a13 + w * a12 + w ^ 2 * a11

def b11Cube : MathValue := reduce9 (b11 * b12 * b13)
def b14Cube : MathValue := reduce9 (b14 * b15 * b16)

(b11Cube, b14Cube, b11Cube * b14Cube)

$(27 w, -27 w - 27, 729)$

Choose cube-root branches¶

The products simplify to $27\omega$ and $27\omega^2$, and their product is $9^3$. We choose conjugate cube-root branches whose product is $9$ and for which the reconstructed period satisfies $a_{11}'>1$. This selects the period $2\cos(2\pi/9)$ uniquely. Taking those roots and applying the inverse Fourier transform gives

$$ a_{11}'=\sqrt[3]{\omega}+\sqrt[3]{\omega^2} =2\cos(2\pi/9). $$

def b11' : MathValue := rt 3 b11Cube
def b14' : MathValue := rt 3 b14Cube

def a11' : MathValue := (b10' + b11' + b14') / 3

a11' / 2

$\frac{1}{2} \sqrt[3]{w} + \frac{1}{2} \sqrt[3]{-w - 1}$

Recover a primitive ninth root¶

As before, $\zeta+\zeta^{-1}=a_{11}'$ turns the last step into the quadratic $x^2-a_{11}'x+1=0$.

def quadraticRoots9
  (a : MathValue)
  (b : MathValue)
  (c : MathValue)
  : (MathValue, MathValue) :=
  ( ((- b) + sqrt (b ^ 2 - 4 * a * c)) / (2 * a)
  , ((- b) - sqrt (b ^ 2 - 4 * a * c)) / (2 * a) )

def z1' : MathValue := fst (quadraticRoots9 1 (- a11') 1)

z1'

$\frac{1}{2} \sqrt{\sqrt[3]{w}^{2} + \sqrt[3]{-w - 1}^{2} + 2 \sqrt[3]{-w - 1} \sqrt[3]{w} - 4} + \frac{1}{2} \sqrt[3]{w} + \frac{1}{2} \sqrt[3]{-w - 1}$

The constructed radical satisfies its final quadratic exactly.

z1' ^ 2 - a11' * z1' + 1

$0$

The cyclotomic check¶

The exact root-of-unity object confirms both the ninth-power relation and its primitive cyclotomic factor.

(z ^ 9, reduce9 (z ^ 6 + z ^ 3 + 1))

$(1, 0)$

Takeaway¶

The factorization of $x^9-1$ isolates a degree-six problem, and the cyclic action on primitive exponents reduces it to cube roots plus one quadratic. The notebook uses only the current typed Egison surface syntax while preserving the original Galois-theoretic idea.