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{
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter 4 - Polynomials"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 121 Example 4.3"
]
},
{
"cell_type": "code",
"execution_count": 1,
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"name": "stdout",
"output_type": "stream",
"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"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 131 Example 4.7"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
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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"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"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 '----------------------------------------------'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 133 Example 4.9"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"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"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"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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