Re: Polynomial GCD and LCM over a Field
- To: mathgroup at smc.vnet.net
- Subject: [mg38514] Re: [mg38507] Polynomial GCD and LCM over a Field
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Fri, 20 Dec 2002 23:40:18 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
On Friday, December 20, 2002, at 06:27 PM, flip wrote:
> Hello,
>
> I have been working with polynomial GCD and LCM over some finite
> fields and
> have a question.
>
> f[x] = x^3 + x + 1
> g[x] = x^2 + 3
> Over F{11}[x] (the field mod 11, forgive my notation).
>
> So, the PolynomialGCD[f[x], g[x], 11] = 1
>
> As a linear combination, we get the GCD as:
>
> 1 = f[x]*a[x] + g[x]*b[x]
> = (x^3 + x + 1)(x+6) + (x^2 + 3)(10x^2 + 5x +2)
>
> No problem, but how does one find the Polynomial LCM?
>
> I assumed it would be: LCM[f[x],g[x]] = (f[x]*g[x])/GCD(f[x],g[x])
>
> so the LCM = (x^3 + x + 1)(x^2 + 3) / 1
> = (x^3 + x + 1)(x^2 + 3)
>
> But, Mathematica shows 11*(x^3 + x + 1)(x^2 + 3). In fact, the 11
> modulo 11
> makes this LCM equal to zero! Can someone explain why?
>
> Thanks for any inputs (several, about 15, books which I looked in
> didn't
> define LCM, they only define GCD).
>
> Flip
>
> P.S. Remove "_alpha" th send me email.
>
>
>
>
>
What do you mean by " Mathematica shows 11*(x^3 + x + 1)(x^2 + 3)"?
Which version of Mathematica? At least 4.2 gives:
In[1]:=
PolynomialLCM[x^3 + x + 1, x^2 + 3, Modulus -> 11]
Out[1]=
(3 + x^2)*(1 + x + x^3)
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/