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MathGroup Archive 2005

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Re: Re: A question about algebraic numbers using Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg62812] Re: [mg62799] Re: [mg62762] A question about algebraic numbers using Mathematica
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Tue, 6 Dec 2005 00:03:08 -0500 (EST)
  • References: <200512050837.DAA08323@smc.vnet.net> <200512051841.NAA21128@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Daniel Lichtblau wrote:
> Kent Holing wrote:
> 
>>I want to find the inverse of 2 - r in Q[r] where r is a root of the equation
>>x^4 - 2c x^3 + (c^2 - 2a^2) x^2 + 2a^2 c x - a^2 c^2 = 0 for a, b and c integers.
>>
>>Can this be done for general a, b and c? (I know how to do it for specific given numerical values of a, b and c.)
>>
>>Kent Holing
> 
> 
> 
> <<Algebra`PolynomialPowerMod`
> 
> InputForm[PolynomialPowerMod[2 - x, -1, x,
>    {x^4 - 2*c*x^3 + (c^2-2*a^2)*x^2 + 2*a^2*c*x - a^2*c^2, 0}]]
> 
> Out[5]//InputForm=
> PolynomialPowerMod[2 - x, -1, x,
>   {-(a^2*c^2) + 2*a^2*c*x + (-2*a^2 + c^2)*x^2 - 2*c*x^3 + x^4, 0}]
> 
> 
> Daniel Lichtblau
> Wolfram Research

Let me try that again.

In[7]:= InputForm[inv = PolynomialPowerMod[2 - x, -1, x,
    {x^4 - 2*c*x^3 + (c^2-2*a^2)*x^2 + 2*a^2*c*x - a^2*c^2, 0}]]

Out[7]//InputForm=
(-8 + 4*a^2 + 8*c - 2*a^2*c - 2*c^2 - 4*x + 2*a^2*x + 4*c*x -
   c^2*x - 2*x^2 + 2*c*x^2 - x^3)/
   (-16 + 8*a^2 + 16*c - 4*a^2*c - 4*c^2 + a^2*c^2)

Let's check the result this time.

In[9]:=  PolynomialMod[inv*(2-x),
   x^4 - 2*c*x^3 + (c^2-2*a^2)*x^2 + 2*a^2*c*x - a^2*c^2]

Out[9]= 1

Daniel Lichtblau
Wolfram Research


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