The 17th Roots of Unity¶

The regular $17$-gon is constructible because $17=2^{2^2}+1$ is a Fermat prime. Algebraically, the Galois group of

$$ \Phi_{17}(x)=x^{16}+x^{15}+\cdots+x+1 $$

has order $16$. A chain of index-two subgroups therefore resolves a primitive root using square roots alone.

Eight real Gaussian periods¶

Let $\zeta=\exp(2\pi i/17)$. Pairing each power with its inverse gives eight real quantities

$$ a_k=\zeta^k+\zeta^{-k}=2\cos(2\pi k/17). $$

def z : MathValue := rtu 17

def a1 : MathValue := z ^ 1 + z ^ 16
def a2 : MathValue := z ^ 2 + z ^ 15
def a3 : MathValue := z ^ 3 + z ^ 14
def a4 : MathValue := z ^ 4 + z ^ 13
def a5 : MathValue := z ^ 5 + z ^ 12
def a6 : MathValue := z ^ 6 + z ^ 11
def a7 : MathValue := z ^ 7 + z ^ 10
def a8 : MathValue := z ^ 8 + z ^ 9

a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8

$-1$

Successive two-point transforms¶

The multiplicative group modulo $17$ is cyclic. The definitions below follow its subgroup chain, repeatedly recording sums and differences. At the top level $d_{10}$ is the sum of all sixteen nontrivial roots and hence equals $-1$.

def b11 : MathValue := a1 + a4
def b12 : MathValue := a1 - a4
def b21 : MathValue := a2 + a8
def b22 : MathValue := a2 - a8
def b31 : MathValue := a3 + a5
def b32 : MathValue := a3 - a5
def b41 : MathValue := a6 + a7
def b42 : MathValue := a6 - a7

def c11 : MathValue := b11 + b21
def c12 : MathValue := b11 - b21
def c21 : MathValue := b31 + b41
def c22 : MathValue := b31 - b41

def d10 : MathValue := c11 + c21
def d11 : MathValue := c11 - c21
def d12 : MathValue := c21 - c11

(d10, d11, d12)

$(-1, -2 rtu(17)^{14} - 2 rtu(17)^{12} - 2 rtu(17)^{11} - 2 rtu(17)^{10} - 2 rtu(17)^{7} - 2 rtu(17)^{6} - 2 rtu(17)^{5} - 2 rtu(17)^{3} - 1, 2 rtu(17)^{14} + 2 rtu(17)^{12} + 2 rtu(17)^{11} + 2 rtu(17)^{10} + 2 rtu(17)^{7} + 2 rtu(17)^{6} + 2 rtu(17)^{5} + 2 rtu(17)^{3} + 1)$

Invert the subgroup chain¶

Each difference has a square that can be expressed using quantities already known one level higher. Choosing compatible square-root branches and repeatedly applying

$$ u=\frac{(u+v)+(u-v)}{2} $$

reconstructs the first period $a_1$.

def d10' : MathValue := -1
def d11' : MathValue := sqrt 17

def c11' : MathValue := (d10' + d11') / 2
def c21' : MathValue := (d10' - d11') / 2
def c12' : MathValue := sqrt (8 + (- c11'))
def c22' : MathValue := sqrt (8 + (- c21'))

def b11' : MathValue := (c11' + c12') / 2
def b21' : MathValue := (c11' - c12') / 2
def b31' : MathValue := (c21' + c22') / 2
def b41' : MathValue := (c21' - c22') / 2

def b12' : MathValue := sqrt (4 + b21' + (-2) * b31')
def b22' : MathValue := sqrt (4 + b21' + (-2) * b41')
def b32' : MathValue := sqrt (4 + b41' + (-2) * b21')
def b42' : MathValue := sqrt (4 + b31' + (-2) * b21')

def a1' : MathValue := (b11' + b12') / 2

Gauss's cosine formula¶

Since $a_1'=2\cos(2\pi/17)$, dividing by two displays the classical nested-square-root construction of the regular $17$-gon:

$$ \cos\frac{2\pi}{17} =\frac1{16}\left( -1+\sqrt{17}+\sqrt{34-2\sqrt{17}} +2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}} -2\sqrt{34+2\sqrt{17}}} \right). $$

a1' / 2

$\frac{1}{16} \sqrt{-\sqrt{17} + 17} \sqrt{2} + \frac{1}{16} \sqrt{17} + \frac{1}{8} \sqrt{-\sqrt{-\sqrt{17} + 17} \sqrt{2} - 2 \sqrt{\sqrt{17} + 17} \sqrt{2} + 3 \sqrt{17} + 17} + \frac{-1}{16}$

Root-of-unity check¶

Egison's exact root-of-unity value retains the defining relation without numerical rounding.

z ^ 17

$1$

Takeaway¶

The nested radicals are a direct trace of four successive index-two reductions in the Galois group. The computation explains not only the value of the cosine but also why straightedge-and- compass construction is possible.