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    "# The 7th Roots of Unity\n",
    "\n",
    "A primitive seventh root $\\zeta$ satisfies\n",
    "\n",
    "$$\n",
    "\\Phi_7(x)=x^6+x^5+x^4+x^3+x^2+x+1=0.\n",
    "$$\n",
    "\n",
    "The six primitive roots are permuted by\n",
    "$(\\mathbb Z/7\\mathbb Z)^\\times$. We first quotient by complex\n",
    "conjugation, leaving three real periods, and then use a\n",
    "three-point Fourier transform over the cube roots of unity.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "046475f3",
   "metadata": {},
   "source": [
    "## Three conjugate periods\n",
    "\n",
    "Pairing exponents $k$ and $7-k$ gives\n",
    "$a_{11},a_{12},a_{13}$. Their sum is $-1$, because the sum of all\n",
    "nontrivial seventh roots is $-1$.\n"
   ]
  },
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   "source": [
    "def z : MathValue := rtu 7\n",
    "\n",
    "def a11 : MathValue := z ^ 1 + z ^ 6\n",
    "def a12 : MathValue := z ^ 2 + z ^ 5\n",
    "def a13 : MathValue := z ^ 3 + z ^ 4\n",
    "\n",
    "def b10 : MathValue := a11 + a12 + a13\n",
    "\n",
    "def cyclotomic7 : MathValue :=\n",
    "  '((rtu 7)^6 + (rtu 7)^5 + (rtu 7)^4\n",
    "    + (rtu 7)^3 + (rtu 7)^2 + rtu 7 + 1)\n",
    "\n",
    "def reduce7 (v : MathValue) : MathValue :=\n",
    "  idealNFWith [w] [cyclotomic7] v\n",
    "\n",
    "def b10' : MathValue := reduce7 b10\n"
   ]
  },
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       "$-1$"
      ]
     },
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    }
   ],
   "source": [
    "b10'\n"
   ]
  },
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   "id": "01802a37",
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   "source": [
    "## Fourier resolvents\n",
    "\n",
    "With $\\omega=(-1+i\\sqrt3)/2$, the two nontrivial characters of the\n",
    "three-cycle produce two conjugate triples. Multiplying each triple\n",
    "removes the cyclic ambiguity. We reduce those products modulo\n",
    "$\\Phi_7(\\zeta)$, leaving expressions in $\\omega$ alone, so one cube\n",
    "root recovers each Fourier component.\n"
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "def b11 : MathValue := a11 + w * a12 + w ^ 2 * a13\n",
    "def b12 : MathValue := a13 + w * a11 + w ^ 2 * a12\n",
    "def b13 : MathValue := a12 + w * a13 + w ^ 2 * a11\n",
    "\n",
    "def b14 : MathValue := a11 + w * a13 + w ^ 2 * a12\n",
    "def b15 : MathValue := a12 + w * a11 + w ^ 2 * a13\n",
    "def b16 : MathValue := a13 + w * a12 + w ^ 2 * a11\n",
    "\n",
    "def b11Cube : MathValue := reduce7 (b11 * b12 * b13)\n",
    "def b14Cube : MathValue := reduce7 (b14 * b15 * b16)\n"
   ]
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   "execution_count": 4,
   "id": "82be293f",
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       "$(21 w + 14, -21 w - 7, 343)$"
      ]
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   ],
   "source": [
    "(b11Cube, b14Cube, b11Cube * b14Cube)\n"
   ]
  },
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   "id": "6a957768",
   "metadata": {},
   "source": [
    "## Invert the transform\n",
    "\n",
    "The two displayed radicands are conjugate and their product is\n",
    "$7^3$. We choose conjugate cube-root branches whose product is $7$\n",
    "and whose reconstructed real period is positive. Their inverse\n",
    "transform gives\n",
    "$a_{11}=\\zeta+\\zeta^{-1}=2\\cos(2\\pi/7)$.\n"
   ]
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  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "bc4cb22a",
   "metadata": {},
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   "source": [
    "def b11' : MathValue := rt 3 b11Cube\n",
    "def b14' : MathValue := rt 3 b14Cube\n",
    "\n",
    "def a11' : MathValue := (b10' + b11' + b14') / 3\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "5f2b3629",
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       "$\\frac{1}{6} \\sqrt[3]{3 w + 2} \\sqrt[3]{7} + \\frac{1}{6} \\sqrt[3]{-3 w - 1} \\sqrt[3]{7} + \\frac{-1}{6}$"
      ]
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   "source": [
    "a11' / 2\n"
   ]
  },
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   "id": "31427058",
   "metadata": {},
   "source": [
    "## Recover a root itself\n",
    "\n",
    "Once $a_{11}'=\\zeta+\\zeta^{-1}$ is known, $\\zeta$ is a root of\n",
    "\n",
    "$$\n",
    "x^2-a_{11}'x+1=0.\n",
    "$$\n",
    "\n",
    "The selected radical branches determine which member of the\n",
    "conjugate pair appears first.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a06ba7dc",
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   "source": [
    "def quadraticRoots7\n",
    "  (a : MathValue)\n",
    "  (b : MathValue)\n",
    "  (c : MathValue)\n",
    "  : (MathValue, MathValue) :=\n",
    "  ( ((- b) + sqrt (b ^ 2 - 4 * a * c)) / (2 * a)\n",
    "  , ((- b) - sqrt (b ^ 2 - 4 * a * c)) / (2 * a) )\n",
    "\n",
    "def z1' : MathValue := fst (quadraticRoots7 1 (- a11') 1)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "8f249803",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\frac{1}{6} \\sqrt[3]{3 w + 2} \\sqrt[3]{7} + \\frac{1}{6} \\sqrt[3]{-3 w - 1} \\sqrt[3]{7} + \\frac{1}{6} \\sqrt{\\sqrt[3]{3 w + 2}^{2} \\sqrt[3]{7}^{2} + \\sqrt[3]{-3 w - 1}^{2} \\sqrt[3]{7}^{2} + 2 \\sqrt[3]{-3 w - 1} \\sqrt[3]{3 w + 2} \\sqrt[3]{7}^{2} - 2 \\sqrt[3]{3 w + 2} \\sqrt[3]{7} - 2 \\sqrt[3]{-3 w - 1} \\sqrt[3]{7} - 35} + \\frac{-1}{6}$"
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   "source": [
    "z1'\n"
   ]
  },
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   "id": "359b8d41",
   "metadata": {},
   "source": [
    "Substitution into the final quadratic gives zero. This check is\n",
    "independent of how the nested square root is formatted.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "5e39bde5",
   "metadata": {},
   "outputs": [
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     "data": {
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       "$0$"
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    }
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   "source": [
    "z1' ^ 2 - a11' * z1' + 1\n"
   ]
  },
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   "id": "3c97d95e",
   "metadata": {},
   "source": [
    "## Takeaway\n",
    "\n",
    "A degree-six cyclotomic equation has been decomposed into one\n",
    "three-point transform, cube roots, and a final quadratic. The code\n",
    "follows the subgroup structure of the Galois group, making the\n",
    "origin of every radical visible.\n"
   ]
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