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Polynomial GCD and LCM over a Field

  • To: mathgroup at smc.vnet.net
  • Subject: [mg38507] Polynomial GCD and LCM over a Field
  • From: "flip" <flip_alpha at safebunch.com>
  • Date: Fri, 20 Dec 2002 04:27:45 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

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.




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