The 7th Roots of Unity¶
A primitive seventh root $\zeta$ satisfies
$$ \Phi_7(x)=x^6+x^5+x^4+x^3+x^2+x+1=0. $$
The six primitive roots are permuted by $(\mathbb Z/7\mathbb Z)^\times$. We first quotient by complex conjugation, leaving three real periods, and then use a three-point Fourier transform over the cube roots of unity.
Three conjugate periods¶
Pairing exponents $k$ and $7-k$ gives $a_{11},a_{12},a_{13}$. Their sum is $-1$, because the sum of all nontrivial seventh roots is $-1$.
def z : MathValue := rtu 7
def a11 : MathValue := z ^ 1 + z ^ 6
def a12 : MathValue := z ^ 2 + z ^ 5
def a13 : MathValue := z ^ 3 + z ^ 4
def b10 : MathValue := a11 + a12 + a13
def cyclotomic7 : MathValue :=
'((rtu 7)^6 + (rtu 7)^5 + (rtu 7)^4
+ (rtu 7)^3 + (rtu 7)^2 + rtu 7 + 1)
def reduce7 (v : MathValue) : MathValue :=
idealNFWith [w] [cyclotomic7] v
def b10' : MathValue := reduce7 b10
b10'
Fourier resolvents¶
With $\omega=(-1+i\sqrt3)/2$, the two nontrivial characters of the three-cycle produce two conjugate triples. Multiplying each triple removes the cyclic ambiguity. We reduce those products modulo $\Phi_7(\zeta)$, leaving expressions in $\omega$ alone, so one cube root recovers each Fourier component.
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 := reduce7 (b11 * b12 * b13)
def b14Cube : MathValue := reduce7 (b14 * b15 * b16)
(b11Cube, b14Cube, b11Cube * b14Cube)
Invert the transform¶
The two displayed radicands are conjugate and their product is $7^3$. We choose conjugate cube-root branches whose product is $7$ and whose reconstructed real period is positive. Their inverse transform gives $a_{11}=\zeta+\zeta^{-1}=2\cos(2\pi/7)$.
def b11' : MathValue := rt 3 b11Cube
def b14' : MathValue := rt 3 b14Cube
def a11' : MathValue := (b10' + b11' + b14') / 3
a11' / 2
Recover a root itself¶
Once $a_{11}'=\zeta+\zeta^{-1}$ is known, $\zeta$ is a root of
$$ x^2-a_{11}'x+1=0. $$
The selected radical branches determine which member of the conjugate pair appears first.
def quadraticRoots7
(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 (quadraticRoots7 1 (- a11') 1)
z1'
Substitution into the final quadratic gives zero. This check is independent of how the nested square root is formatted.
z1' ^ 2 - a11' * z1' + 1
Takeaway¶
A degree-six cyclotomic equation has been decomposed into one three-point transform, cube roots, and a final quadratic. The code follows the subgroup structure of the Galois group, making the origin of every radical visible.