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   "source": [
    "# 1の5乗根\n",
    "\n",
    "$\\zeta=\\exp(2\\pi i/5)$ とおきます。1の原始5乗根は、次の方程式を\n",
    "満たします。\n",
    "\n",
    "$$\n",
    "\\Phi_5(x)=x^4+x^3+x^2+x+1=0.\n",
    "$$\n",
    "\n",
    "特に、\n",
    "\n",
    "$$\n",
    "\\cos\\frac{2\\pi}{5}=\\frac{\\sqrt5-1}{4}.\n",
    "$$\n",
    "\n",
    "この冪根表示を導く際には、ブラックボックスの多項式ソルバーに四つの解を一度に\n",
    "求めさせるのではなく、$\\zeta$ の冪を軌道ごとに整理します。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1a99d81b",
   "metadata": {},
   "source": [
    "## 共役な冪を組にする\n",
    "\n",
    "複素共役により、$\\zeta$ と $\\zeta^4$、$\\zeta^2$ と $\\zeta^3$ がそれぞれ\n",
    "対になります。各対の対称和は実数です。この二つの対を足すと、すべての原始5乗根の\n",
    "和である $-1$ が得られます。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "135603da",
   "metadata": {},
   "outputs": [],
   "source": [
    "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "dfd4fa16",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$(-1, -2 rtu(5)^{3} - 2 rtu(5)^{2} - 1, 2 rtu(5)^{3} + 2 rtu(5)^{2} + 1)$"
      ]
     },
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   ],
   "source": [
    "(b10, b11, b12)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "886e1b22",
   "metadata": {},
   "source": [
    "## 最初の2点変換を逆変換する\n",
    "\n",
    "和はすでに $b_{10}=-1$ と分かっています。差を二乗すれば符号の曖昧さが消えます。\n",
    "平方根の分枝を選んで逆変換を適用すると、二つの実周期を再構成できます。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "d87f029a",
   "metadata": {},
   "outputs": [],
   "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "4e6d75ed",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$(\\frac{1}{2} \\sqrt{5} + \\frac{-1}{2}, \\frac{-1}{2} \\sqrt{5} + \\frac{-1}{2})$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "(a11', a12')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "33337db7",
   "metadata": {},
   "source": [
    "## 虚部を復元する\n",
    "\n",
    "反対称な対 $\\zeta-\\zeta^{-1}$ と $\\zeta^2-\\zeta^{-2}$ が虚部を担います。\n",
    "2回目の2点変換により、これらは平方根へ還元されます。その被開平数は、先ほど\n",
    "求めた実周期だけに依存します。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "d92e73c6",
   "metadata": {},
   "outputs": [],
   "source": [
    "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "1b8998bb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\frac{1}{4} \\sqrt{2 \\sqrt{5} + 5} i + \\frac{1}{4} \\sqrt{-2 \\sqrt{5} + 5} i + \\frac{1}{4} \\sqrt{5} + \\frac{-1}{4}$"
      ]
     },
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    }
   ],
   "source": [
    "z1'\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "480967c9",
   "metadata": {},
   "source": [
    "## 冪根の関係を検証する\n",
    "\n",
    "入れ子になった平方根には、代数的な関係をもつ、分枝を表す記号が導入されます。\n",
    "次のイデアルに、それらの定義関係を記録します。そのイデアルを法として\n",
    "$(z_1')^5-1$ を簡約すれば、浮動小数点による近似ではなく、厳密な記号計算として\n",
    "検証できます。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "7e6b81b5",
   "metadata": {},
   "outputs": [],
   "source": [
    "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "16e08df0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$0$"
      ]
     },
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     "output_type": "display_data"
    }
   ],
   "source": [
    "idealNF radicalRels (z1' ^ 5 - 1)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b138b53c",
   "metadata": {},
   "source": [
    "## まとめ\n",
    "\n",
    "黄金比でおなじみの平方根が現れるのは、$\\Phi_5$ のガロア群を二段階の2点変換で\n",
    "解けるためです。Egisonでは、軌道和の計算と冪根表示の厳密な検証を、同じ記号計算の\n",
    "中で扱えます。\n"
   ]
  }
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