{
 "cells": [
  {
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   "id": "10597d44",
   "metadata": {},
   "source": [
    "MCS 320 Quiz 2 Friday 21 June 2024"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6714c80f",
   "metadata": {},
   "source": [
    "# Question 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "203118a7-77fd-4a26-98cf-8c75b4589994",
   "metadata": {},
   "source": [
    "Compute the polar representation of the complex number $7 - 4 I$.\n",
    "\n",
    "Verify that the polar representation you computed corresponds to $7 - 4 I$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "117c493c-f953-4856-8788-239c9e5ea559",
   "metadata": {},
   "source": [
    "## answer to question 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "0659b309-50c1-43f3-b90d-09695b159da2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-4*I + 7"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = 7 - 4*I\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "273a02f3-9d07-4596-a7bd-fc425792f0b9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sqrt(65)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = abs(z)\n",
    "r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "4283be30-0807-48db-b3f7-618aae73a777",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-arctan(4/7)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = arg(z)\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b73cac89-57c7-40a4-b3f9-7705374f87e8",
   "metadata": {},
   "source": [
    "The polar representation of `z` is `r*exp(I*a)` computed below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "08affda4-e5ef-4286-a7e7-cd6036b92acb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sqrt(65)*e^(-I*arctan(4/7))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pz = r*exp(I*a)\n",
    "pz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "b9e1301c-0622-4299-aab1-a34b43e29fba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rz = simplify(pz.real())\n",
    "rz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "8d5e38eb-7213-46f5-89fa-09be708f5fee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-4"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "iz = simplify(pz.imag())\n",
    "iz"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0567ca7c-d842-4fe7-b7e6-a5ae23aaaa52",
   "metadata": {},
   "source": [
    "Thus we see that the real part and the imaginary part of the polar representation `pz` simplifies to"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a0b654cb-777b-4e58-97fc-6ed164100319",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-4*I + 7"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rz + iz*I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "6004628b-eb64-4888-83d7-a06f9c9b3c64",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-4*I + 7"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85378d99-9980-489a-a6e8-7c4d9d6b6c03",
   "metadata": {},
   "source": [
    "which indeed equals `z`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd6eb31f",
   "metadata": {},
   "source": [
    "# Question 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b365ff05",
   "metadata": {},
   "source": [
    "Consider the polynomial $p = x^3 + 8 x + 7$ and compute modulo 13.\n",
    "\n",
    "1. Define $p$ as a polynomial in $x$ with modulo 13 coefficients.\n",
    "   Does $p$ factor?\n",
    "\n",
    "    \n",
    "2. Let $\\alpha$ be the formal root of $p$ used in the field extension\n",
    "   of the modulo 13 numbers.\n",
    "    \n",
    "   How many elements are in this extended finite field?\n",
    "   Write $\\alpha^7$ in its normal form."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "039a94a7",
   "metadata": {},
   "source": [
    "## answer to question 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "cfda5db0",
   "metadata": {},
   "outputs": [],
   "source": [
    "reset()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "fb3e1a20",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x^3 + 8*x + 7 has type <class 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>\n"
     ]
    }
   ],
   "source": [
    "x = polygen(GF(13))\n",
    "p = x^3 + 8*x + 7\n",
    "print(p, 'has type', type(p))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "dd718a54",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.is_irreducible()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "5cdad283",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finite Field in alpha of size 13^3"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.<alpha> = GF(13).extension(p)\n",
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "28766f78",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2197"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "13^3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f4a06ff",
   "metadata": {},
   "source": [
    "There are 2197 elements in the extension field."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "f9351c7f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8*alpha^2 + 5*alpha + 7"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha^7"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9e70e2b7",
   "metadata": {},
   "source": [
    "The representation of $\\alpha^7$ is $8 \\alpha^2 + 5 \\alpha + 7$."
   ]
  }
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