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   "source": [
    "# Chapter 4 - Polynomials"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 121 Example 4.3"
   ]
  },
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     "text": [
      "C is the field of complex numbers\n",
      "f =  x**2 + 2\n",
      "if a = C and z belongs to C, then f(z) = z**2 + 2\n",
      "f(2) =  6\n",
      "f(1+1J/1-1J) =  (1+0j)\n",
      "----------------------------------------\n",
      "If a is the algebra of all 2*2 matrices over C and\n",
      "B = \n",
      "[[ 1  0]\n",
      " [-1  2]]\n",
      "[[ 3.  0.]\n",
      " [ 1.  6.]] then, f(B) = \n",
      "----------------------------------------\n",
      "If a is algebra of all linear operators on C**3\n",
      "And T is element of a as:\n",
      "T(c1,c2,c3) = (i*2**1/2*c1,c2,i*2**1/2*c3)\n",
      "Then, f(T)(c1,c2,c3) = (0,3*c2,0)\n",
      "----------------------------------------\n",
      "If a is the algebra of all polynomials over C\n",
      "And, g =  x**4 + 3.0*I\n",
      "Then f(g) =  (16+3j)\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import sympy as sp\n",
    "from sympy.polys.polyfuncs import horner\n",
    "print 'C is the field of complex numbers'\n",
    "x = sp.Symbol(\"x\")\n",
    "def f(x):\n",
    "    ff= x**2 + 2\n",
    "    return ff\n",
    "print 'f = ',f(x)\n",
    "#part a\n",
    "print 'if a = C and z belongs to C, then f(z) = z**2 + 2'\n",
    "print 'f(2) = ',f(2)\n",
    "\n",
    "\n",
    "print 'f(1+1J/1-1J) = ',f((1+1J)/(1-1J))\n",
    "print '----------------------------------------'\n",
    "\n",
    "\n",
    "#part b\n",
    "print 'If a is the algebra of all 2*2 matrices over C and'\n",
    "B = np.array([[1 ,0],[-1, 2]])\n",
    "print 'B = \\n',B\n",
    "print 2*np.identity(2) + B**2,'then, f(B) = '\n",
    "print '----------------------------------------'\n",
    "\n",
    "#part c\n",
    "print 'If a is algebra of all linear operators on C**3'\n",
    "print 'And T is element of a as:'\n",
    "print 'T(c1,c2,c3) = (i*2**1/2*c1,c2,i*2**1/2*c3)'\n",
    "print 'Then, f(T)(c1,c2,c3) = (0,3*c2,0)'\n",
    "print '----------------------------------------'\n",
    "#part d\n",
    "print 'If a is the algebra of all polynomials over C'\n",
    "def g(x):\n",
    "    gg= x**4 + 3*1J\n",
    "    return gg\n",
    "print 'And, g = ',g(x)\n",
    "print 'Then f(g) = ',g(2)\n"
   ]
  },
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   "metadata": {},
   "source": [
    "## Page 131 Example 4.7"
   ]
  },
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M = (x+2)F[x] + (x**2 + 8x + 16)F[x]\n",
      "We assert, M = F[x]\n",
      "M contains: x**2 - x*(x + 2) + 8*x + 16\n",
      "And hence M contains: x**2 - x*(x + 2) + 2*x + 4\n",
      "Thus the scalar polynomial 1 belongs to M as well all its multiples.\n"
     ]
    }
   ],
   "source": [
    "import sympy as sp\n",
    "x = sp.Symbol('x')\n",
    "p1 = x + 2#\n",
    "p2 = x**2 + 8*x + 16#\n",
    "print 'M = (x+2)F[x] + (x**2 + 8x + 16)F[x]'\n",
    "print 'We assert, M = F[x]'\n",
    "print 'M contains:',\n",
    "t = p2 - x*p1#\n",
    "print t\n",
    "print 'And hence M contains:',\n",
    "print t - 6*p1\n",
    "print 'Thus the scalar polynomial 1 belongs to M as well all its multiples.'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 133 Example 4.8"
   ]
  },
  {
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   "execution_count": 3,
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   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "p1 =  x + 2\n",
      "p2 =  x**2 + 8*x + 16\n",
      "M = (x+2)F[x] + (x**2 + 8x + 16)F[x]\n",
      "We assert, M = F[x]\n",
      "M contains: x**2 - x*(x + 2) + 8*x + 16\n",
      "And hence M contains: x**2 - x*(x + 2) + 2*x + 4\n",
      "Thus the scalar polynomial 1 belongs to M as well all its multiples\n",
      "So, gcd(p1,p2) = 1\n",
      "----------------------------------------------\n",
      "p1 =  (x - 2)**2*(x + 1.0*I)\n",
      "p2 =  (x - 2)*(x**2 + 1)\n",
      "M = (x - 2)**2*(x+1J)F[x] + (x-2)*(x**2 + 1\n",
      "The ideal M contains p1 - p2 i.e., (x - 2)**2*(x + 1.0*I) - (x - 2)*(x**2 + 1)\n",
      "Hence it contains (x-2)(x+i), which is monic and divides both,\n",
      "So, gcd(p1,p2) = (x-2)(x+i)\n",
      "----------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "import sympy as sp\n",
    "x = sp.Symbol('x')\n",
    "\n",
    "#part a\n",
    "p1 = x + 2#\n",
    "p2 = x**2 + 8*x + 16#\n",
    "print 'p1 = ',p1\n",
    "print 'p2 = ',p2\n",
    "print 'M = (x+2)F[x] + (x**2 + 8x + 16)F[x]'\n",
    "print 'We assert, M = F[x]'\n",
    "print 'M contains:',\n",
    "t = p2 - x*p1#\n",
    "print t\n",
    "print 'And hence M contains:',\n",
    "print t - 6*p1\n",
    "print 'Thus the scalar polynomial 1 belongs to M as well all its multiples'\n",
    "print 'So, gcd(p1,p2) = 1'\n",
    "print '----------------------------------------------'\n",
    "#part b\n",
    "p1 = (x - 2)**2*(x+1J)#\n",
    "p2 = (x-2)*(x**2 + 1)#\n",
    "print 'p1 = ',p1\n",
    "print 'p2 = ',p2\n",
    "print 'M = (x - 2)**2*(x+1J)F[x] + (x-2)*(x**2 + 1'\n",
    "print 'The ideal M contains p1 - p2 i.e.,',\n",
    "print p1 - p2\n",
    "print 'Hence it contains (x-2)(x+i), which is monic and divides both,'\n",
    "print 'So, gcd(p1,p2) = (x-2)(x+i)'\n",
    "print '----------------------------------------------'"
   ]
  },
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   "metadata": {},
   "source": [
    "## Page 133 Example 4.9"
   ]
  },
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   "execution_count": 4,
   "metadata": {
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   "outputs": [
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     "text": [
      "M is the ideal in F[x] generated by:\n",
      "(x-1)*(x+2)**2\n",
      "(x+2)**2*(x+3)\n",
      "(x-3) and\n",
      "M = (x-1)*(x+2)**2 F[x] + (x+2)**2*(x-3) + (x-3)\n",
      "Then M contains: 0\n",
      "i.e., M contains (x+2)**2\n",
      "and since, (x+2)**2 = (x-3)(x-7) - 17\n",
      "So M contains the scalar polynomial 1.\n",
      "So, M = F[x] and given polynomials are relatively prime.\n"
     ]
    }
   ],
   "source": [
    "import sympy as sp\n",
    "x = sp.Symbol('x')\n",
    "\n",
    "print 'M is the ideal in F[x] generated by:'\n",
    "print '(x-1)*(x+2)**2'\n",
    "print '(x+2)**2*(x+3)'\n",
    "print '(x-3)','and'\n",
    "p1 = (x-1)*(x+2)**2#\n",
    "p2 = (x+2)**2*(x-3)#\n",
    "p3 = (x-3)#\n",
    "print 'M = (x-1)*(x+2)**2 F[x] + (x+2)**2*(x-3) + (x-3)'\n",
    "print 'Then M contains:',\n",
    "t = 1/2*(x+2)**2*((x-1) - (x-3))#\n",
    "print t\n",
    "print 'i.e., M contains (x+2)**2'\n",
    "print 'and since, (x+2)**2 = (x-3)(x-7) - 17'\n",
    "print 'So M contains the scalar polynomial 1.'\n",
    "print 'So, M = F[x] and given polynomials are relatively prime.'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 135 Example 4.10"
   ]
  },
  {
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   "execution_count": 5,
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x**2 + 1 P = \n",
      "P is reducible over complex numbers as:  = x**2 + 1\n",
      "(x-i)(x+i)\n",
      "Whereas, P is irreducible over real numbers as: = x**2 + 1\n",
      "(ax + b)(ax + b)\n",
      "For, a,a,b,b to be in R,\n",
      "aa = 1\n",
      "ab + ba = 0\n",
      "bb = 1\n",
      "=> a**2 + b**2 = 0\n",
      "=> a = b = 0\n"
     ]
    }
   ],
   "source": [
    "import sympy as sp\n",
    "x = sp.Symbol('x')\n",
    "P = x**2 + 1#\n",
    "print P,'P = '\n",
    "print 'P is reducible over complex numbers as: ',\n",
    "print '=',P\n",
    "print '(x-i)(x+i)'\n",
    "print 'Whereas, P is irreducible over real numbers as:',\n",
    "print '=',P\n",
    "print '(ax + b)(a''x + b'')'\n",
    "print 'For, a,a'',b,b'' to be in R,'\n",
    "print 'aa'' = 1'\n",
    "print 'ab'' + ba'' = 0'\n",
    "print 'bb'' = 1'\n",
    "print '=> a**2 + b**2 = 0'\n",
    "print '=> a = b = 0'"
   ]
  }
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