{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "fe8e36b9-e53e-4d98-9e58-172fbc2a95a0",
   "metadata": {},
   "source": [
    "最初にヤコビ記号がかっこよく表示されるように指示します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "4cc05c5b-2480-4d3d-9d98-ee21bb5c1697",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "texput(jacobi,lambda([f],concat(\"\\\\left(\\\\frac{\",tex1(args(f)[1]),\"}{\",tex1(args(f)[2]),\"}\\\\right)\")))$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8c693bea-febb-4c5e-9ae3-48f71996078d",
   "metadata": {},
   "source": [
    "次に命題4.11にしたがって幾つかの関数を定義します。\n",
    "まず$qast(q)$を定義します。これは$q^*$の定義になっています。\n",
    "次に$jcbast(q,p)$を定義します。$jcbast(q,p)$は２つの素数$q,p$から$\\left(\\frac{\\left(\\frac{-1}{q}\\right)\\,q}{p}\\right)$を計算します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "cea73f01-d9a7-4af9-9411-b1dda6355430",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "qast(q):=jacobi(-1,q)*q$\n",
    "jcbast(q,p):=jacobi(qast(q),p)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c0a57b2a-d3d4-4a6a-bb06-0fcc1b2d6d8f",
   "metadata": {
    "tags": []
   },
   "source": [
    "以下の式が成り立つことを幾つかの具体的な素数の組で確認します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "1ca6ab5e-1e38-40db-a23b-d847ef0c66a5",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{7}$}\\left(\\frac{\\left(\\frac{-1}{q}\\right)\\,q}{p}\\right)=\\left(\\frac{p}{q}\\right)\\]"
      ],
      "text/plain": [
       "(%o7)             jacobi(jacobi(- 1, q) q, p) = jacobi(p, q)"
      ],
      "text/x-maxima": [
       "jacobi(jacobi(-1,q)*q,p) = jacobi(p,q)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jcbast(q,p)=jacobi(p,q);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d28be4ad-a3af-4f6c-8c34-052ecfbd18bb",
   "metadata": {
    "tags": []
   },
   "source": [
    "こんなコードを走らせるとどの素数の組でも値が同じになることが分かります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "a65dfb8c-cd82-4be7-afc0-98c094867b5a",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{8}$}\\left[ \\left[ 1 , 1 \\right]  , \\left[ -1 , -1 \\right]  , \\left[ -1 , -1 \\right]  , \\left[ -1 , -1 \\right]  , \\left[ 1 , 1 \\right]  \\right] \\]"
      ],
      "text/plain": [
       "(%o8)        [[1, 1], [- 1, - 1], [- 1, - 1], [- 1, - 1], [1, 1]]"
      ],
      "text/x-maxima": [
       "[[1,1],[-1,-1],[-1,-1],[-1,-1],[1,1]]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "map(lambda([arg],block([p,q],[p,q]:arg,[jcbast(p,q),jacobi(q,p)])),\n",
    "    [[3,7],[11,13],[37,19],[31,91],[53,11]]);"
   ]
  }
 ],
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