{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "673d534f-4cde-40bf-82c6-24b0e78fa1c8",
   "metadata": {},
   "source": [
    "maximaを起動したらまず以下のコマンドでGF.macを読み込みます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "6eda2588-8023-41ea-97e5-ccd8e30734ba",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{0}$}\\mbox{ /opt/homebrew/Cellar/maxima/5.46.0\\_10/share/maxima/5.46.0/share/contrib/gf/GF.mac }\\]"
      ],
      "text/plain": [
       "(%o0) /opt/homebrew/Cellar/maxima/5.46.0_10/share/maxima/5.46.0/share/contrib/\\\n",
       "gf/GF.mac"
      ],
      "text/x-maxima": [
       "\"/opt/homebrew/Cellar/maxima/5.46.0_10/share/maxima/5.46.0/share/contrib/gf/GF.mac\""
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "load(GF);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f6d0ce8a-a77d-4397-88fe-b514991b5eb1",
   "metadata": {},
   "source": [
    "まず$F_5$には$\\sqrt{2}$は存在しないことを確認しておきます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "baeaec8c-a0d9-4721-96cc-ec0c4dd8d310",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{1}$}-1\\]"
      ],
      "text/plain": [
       "(%o1)                                 - 1"
      ],
      "text/x-maxima": [
       "-1"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jacobi(2,5);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e1904f0f-d525-4fdb-94f8-313997145bcd",
   "metadata": {},
   "source": [
    "$F_5(\\sqrt{2})$を構成します。gf_xxxコマンドはこの体の上で動作します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "00ee38d7-92c4-4650-b9b3-cb25cea1ab87",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{2}$}\\mbox{ Structure [GF-DATA] }\\]"
      ],
      "text/plain": [
       "(%o2)                         Structure [GF-DATA]"
      ],
      "text/x-maxima": [
       "Structure\\ \\[GF\\-DATA\\]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gf_set_data(5,x^2-2);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "372132d3-48cd-4c56-b0cc-8368491fb901",
   "metadata": {},
   "source": [
    "これで$F_5(\\sqrt{2})$が構成できました。以下で登場する$x$は$\\sqrt{2}$を指すと考えてください。例えば$x$に$x$をかけると$2$になるはずです。乗算にはgf_multを使います。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "a4940084-c7a7-4bde-801e-6b1709f865be",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{3}$}2\\]"
      ],
      "text/plain": [
       "(%o3)                                  2"
      ],
      "text/x-maxima": [
       "2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gf_mult(x,x);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2b52bbd-783c-4c3b-9c5a-7031b7da6090",
   "metadata": {},
   "source": [
    "$F_5(\\sqrt{2})^{\\times}$は巡回群になるのですが、gf_primitiveを使ってその生成元を求めることもできます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "1aeac752-0d91-4893-8ccc-3f6a53047656",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{4}$}x+2\\]"
      ],
      "text/plain": [
       "(%o4)                                x + 2"
      ],
      "text/x-maxima": [
       "x+2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a:gf_primitive();"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "41041b89-9a0d-40df-a077-a485c4f06d48",
   "metadata": {},
   "source": [
    "これは$\\sqrt{2}+2$が生成元だということです。実際に$x+2$の冪乗をいくつか求めてみます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "45c2282b-2c25-4d31-b208-1996539054b4",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{5}$}\\left[ 1 , x+2 , 4\\,x+1 , 4\\,x , 3\\,x+3 , 4\\,x+2 , 3\\,x+1 , 2\\,x+3 , 2\\,x , 4\\,x+4 , 2\\,x+1 , 1 \\right] \\]"
      ],
      "text/plain": [
       "(%o5) [1, x + 2, 4 x + 1, 4 x, 3 x + 3, 4 x + 2, 3 x + 1, 2 x + 3, 2 x, \n",
       "                                                           4 x + 4, 2 x + 1, 1]"
      ],
      "text/x-maxima": [
       "[1,x+2,4*x+1,4*x,3*x+3,4*x+2,3*x+1,2*x+3,2*x,4*x+4,2*x+1,1]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "map(lambda([i],gf_exp(x+2,i)),[0,1,2,3,4,5,19,20,21,22,23,24]);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f329036-d8c9-490b-b41c-7c8c22906af1",
   "metadata": {},
   "source": [
    "**命題3.34**の公式 $(a+b)^p=a^p+b^p$を実際の値を入れて試してみます。$F_5(\\sqrt{2})$で$(a+b)^5=a^5+b^5$を試してみましょう。適当に$a=\\sqrt{2}+3,\\, b=3\\sqrt{2}+1$とおいてみます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "7b698aa5-da7e-44ba-a2ac-ec8c20307eb4",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{6}$}x+3\\]"
      ],
      "text/plain": [
       "(%o6)                                x + 3"
      ],
      "text/x-maxima": [
       "x+3"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{7}$}3\\,x+1\\]"
      ],
      "text/plain": [
       "(%o7)                               3 x + 1"
      ],
      "text/x-maxima": [
       "3*x+1"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a:x+3;b:3*x+1;"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b3c6d375-6e84-4c93-9a86-587bab6d222e",
   "metadata": {},
   "source": [
    "まず$(a+b)^5$を計算します。gf_addで加算、gf_expで冪乗を計算します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "f0fc580d-34a7-42bc-bc03-e5caf14ad935",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{8}$}x+4\\]"
      ],
      "text/plain": [
       "(%o8)                                x + 4"
      ],
      "text/x-maxima": [
       "x+4"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gf_exp(gf_add(a,b),5);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "358a13f1-bbce-477c-bb17-62082a006fba",
   "metadata": {},
   "source": [
    "結果は$\\sqrt{2}+4$です。\n",
    "今度は$a^5+b^5$を計算してみます。先にgf_expで冪乗を求めてからgf_addで加算します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "8601a4ea-455c-493e-821c-04f4a19c6dd9",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{9}$}x+4\\]"
      ],
      "text/plain": [
       "(%o9)                                x + 4"
      ],
      "text/x-maxima": [
       "x+4"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gf_add(gf_exp(a,5),gf_exp(b,5));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dcab666c-88e5-4f5f-bcb6-26bc61ee57cb",
   "metadata": {},
   "source": [
    "結果は$\\sqrt{2}+4$で同じ値になりました。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "125a5996-831f-47a4-8ed5-ffe43282b9a6",
   "metadata": {
    "tags": []
   },
   "source": [
    "次は**命題3.48**が$F_p$で成り立つこと、つまり$a\\in　F_p \\iff a^p=a$を値を入れて試してみます。まず任意の$F_5$の元$c$について$c^5=c$が成り立つことを確認します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "4eed2712-a1b2-4ebb-974c-5ed32e833dcd",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{10}$}\\left[ \\left[ 1 , 1 \\right]  , \\left[ 2 , 2 \\right]  , \\left[ 3 , 3 \\right]  , \\left[ 4 , 4 \\right]  \\right] \\]"
      ],
      "text/plain": [
       "(%o10)                 [[1, 1], [2, 2], [3, 3], [4, 4]]"
      ],
      "text/x-maxima": [
       "[[1,1],[2,2],[3,3],[4,4]]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "map(lambda([c],[c,gf_exp(c,5)]),[1,2,3,4]);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e34523f2-58c1-4c88-a4c7-3eff13ed2d92",
   "metadata": {},
   "source": [
    "$F_5$の元ではない場合についても確認します。先ほど定義した$a,b$はいずれも$F_5$の元ではありませんから、$a^5\\neq a,\\, b^5\\neq b$のはずです。やってみましょう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "57f4dc3f-210b-4798-a37e-45431d27036d",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{11}$}\\left[ x+3 , 4\\,x+3 \\right] \\]"
      ],
      "text/plain": [
       "(%o11)                         [x + 3, 4 x + 3]"
      ],
      "text/x-maxima": [
       "[x+3,4*x+3]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{12}$}\\left[ 3\\,x+1 , 2\\,x+1 \\right] \\]"
      ],
      "text/plain": [
       "(%o12)                        [3 x + 1, 2 x + 1]"
      ],
      "text/x-maxima": [
       "[3*x+1,2*x+1]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[a,gf_exp(a,5)];\n",
    "[b,gf_exp(b,5)];"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb53cd5e-1813-47a0-a21e-a112a4b74a22",
   "metadata": {
    "tags": []
   },
   "source": [
    "**命題3.48**が$F_{p^f}$で成り立つこと、つまり$a\\in F_{p^f} \\iff a^{p^f}=a$を値を入れて試してみます。$F_{5^2}=F_{25}$なので$F_5(\\sqrt{2})$の任意の要素$x$で$x^{25}=x$が成り立ちます。先ほど定義した$a,b$でも成り立つことを見ましょう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "83e8159a-0bfd-4fde-a35f-e0d2b4150149",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{13}$}\\left[ x+3 , x+3 \\right] \\]"
      ],
      "text/plain": [
       "(%o13)                          [x + 3, x + 3]"
      ],
      "text/x-maxima": [
       "[x+3,x+3]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[a,gf_exp(a,25)];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "28d57048-6338-4cd0-82ee-09e2a503f8dd",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{14}$}\\left[ 3\\,x+1 , 3\\,x+1 \\right] \\]"
      ],
      "text/plain": [
       "(%o14)                        [3 x + 1, 3 x + 1]"
      ],
      "text/x-maxima": [
       "[3*x+1,3*x+1]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[b,gf_exp(b,25)];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "062cf663-312c-4ad9-9f89-e002d8734ec6",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Maxima",
   "language": "maxima",
   "name": "maxima"
  },
  "language_info": {
   "codemirror_mode": "maxima",
   "file_extension": ".mac",
   "mimetype": "text/x-maxima",
   "name": "maxima",
   "pygments_lexer": "maxima",
   "version": "5.46.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}