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    "# The 5th Roots of Unity\n",
    "\n",
    "Let $\\zeta=\\exp(2\\pi i/5)$. The primitive fifth roots satisfy\n",
    "\n",
    "$$\n",
    "\\Phi_5(x)=x^4+x^3+x^2+x+1=0.\n",
    "$$\n",
    "\n",
    "In particular,\n",
    "\n",
    "$$\n",
    "\\cos\\frac{2\\pi}{5}=\\frac{\\sqrt5-1}{4}.\n",
    "$$\n",
    "\n",
    "We recover this radical expression by organizing powers of\n",
    "$\\zeta$ into orbits rather than asking a black-box polynomial\n",
    "solver for four roots at once.\n"
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   "source": [
    "## Pair conjugate powers\n",
    "\n",
    "Complex conjugation pairs $\\zeta$ with $\\zeta^4$ and $\\zeta^2$\n",
    "with $\\zeta^3$. Their symmetric sums are real. Adding the two\n",
    "pairs gives $-1$, the sum of all primitive fifth roots.\n"
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    "def z : MathValue := rtu 5\n",
    "\n",
    "def a11 : MathValue := z ^ 1 + z ^ 4\n",
    "def a12 : MathValue := z ^ 2 + z ^ 3\n",
    "\n",
    "def b10 : MathValue := a11 + a12\n",
    "def b11 : MathValue := a11 - a12\n",
    "def b12 : MathValue := a12 - a11\n"
   ]
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       "$(-1, -2 rtu(5)^{3} - 2 rtu(5)^{2} - 1, 2 rtu(5)^{3} + 2 rtu(5)^{2} + 1)$"
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   "source": [
    "(b10, b11, b12)\n"
   ]
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   "source": [
    "## Invert the first two-point transform\n",
    "\n",
    "The sum is already $b_{10}=-1$. Squaring the difference removes\n",
    "its sign ambiguity; choosing a square-root branch and applying the\n",
    "inverse transform reconstructs the two real periods.\n"
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   "source": [
    "def b10' : MathValue := b10\n",
    "def b11' : MathValue := sqrt (b11 ^ 2)\n",
    "\n",
    "def a11' : MathValue := (b10' + b11') / 2\n",
    "def a12' : MathValue := (b10' - b11') / 2\n"
   ]
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       "$(\\frac{1}{2} \\sqrt{5} + \\frac{-1}{2}, \\frac{-1}{2} \\sqrt{5} + \\frac{-1}{2})$"
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    "(a11', a12')\n"
   ]
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   "source": [
    "## Recover the imaginary parts\n",
    "\n",
    "The antisymmetric pairs\n",
    "$\\zeta-\\zeta^{-1}$ and $\\zeta^2-\\zeta^{-2}$ carry the imaginary\n",
    "parts. A second two-point transform reduces them to square roots\n",
    "whose radicands depend only on the real periods above.\n"
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    "def a21 : MathValue := z ^ 1 - z ^ 4\n",
    "def a22 : MathValue := z ^ 2 - z ^ 3\n",
    "\n",
    "def b20 : MathValue := a21 + a22\n",
    "def b21 : MathValue := a21 - a22\n",
    "def b22 : MathValue := a22 - a21\n",
    "\n",
    "def b20' : MathValue := sqrt ((-3) + 4 * a12')\n",
    "def b21' : MathValue := sqrt ((-3) + 4 * a11')\n",
    "\n",
    "def a21' : MathValue := (b20' + b21') / 2\n",
    "def a22' : MathValue := (b20' - b21') / 2\n",
    "\n",
    "def z1' : MathValue := (a11' + a21') / 2\n"
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       "$\\frac{1}{4} \\sqrt{2 \\sqrt{5} + 5} i + \\frac{1}{4} \\sqrt{-2 \\sqrt{5} + 5} i + \\frac{1}{4} \\sqrt{5} + \\frac{-1}{4}$"
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    "z1'\n"
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   "source": [
    "## Verify the radical relation\n",
    "\n",
    "Nested square roots introduce branch atoms that are algebraically\n",
    "related. The following ideal records their defining relations.\n",
    "Reducing $(z_1')^5-1$ modulo that ideal gives an exact symbolic\n",
    "check, rather than a floating-point approximation.\n"
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    "def radicalRels : [MathValue] :=\n",
    "  [ '((sqrt (5 + 2 * sqrt 5)) ^ 2 - 5 - 2 * sqrt 5)\n",
    "  , '((sqrt (5 - 2 * sqrt 5)) ^ 2 - 5 + 2 * sqrt 5)\n",
    "  , '((sqrt (5 + 2 * sqrt 5))\n",
    "        * (sqrt (5 - 2 * sqrt 5))\n",
    "        - sqrt 5)\n",
    "  , '((sqrt 5) ^ 2 - 5)\n",
    "  , '(i ^ 2 + 1) ]\n"
   ]
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   "source": [
    "idealNF radicalRels (z1' ^ 5 - 1)\n"
   ]
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   "source": [
    "## Takeaway\n",
    "\n",
    "The familiar golden-ratio square root appears because the Galois\n",
    "group of $\\Phi_5$ can be resolved by two successive two-point\n",
    "transforms. Egison keeps the orbit sums and the exact radical\n",
    "verification in the same symbolic calculation.\n"
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