Re: Polynomial Reduction with Mod

*To*: mathgroup at smc.vnet.net*Subject*: [mg30440] Re: [mg30436] Polynomial Reduction with Mod*From*: Ken Levasseur <Kenneth_Levasseur at uml.edu>*Date*: Sun, 19 Aug 2001 02:01:34 -0400 (EDT)*References*: <200108180804.EAA22597@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Flip: If you want to do something involving polynomials and don't know the name of the function that might do it, try doing this. In[1]:= ?*Polynomial* CharacteristicPolynomial PolynomialQ InterpolatingPolynomial PolynomialQuotient PolynomialForm PolynomialReduce PolynomialGCD PolynomialRemainder PolynomialLCM Polynomials PolynomialMod You might think that it's PolynomialMod that does what you want, but since modular calculations are remainders, PolynomialRemainder is the one. You would see that by executing ?PolynomialMod you get a different result. In[2]:= ?PolynomialRemainder \!\(TraditionalForm\`"PolynomialRemainder[p, q, x] gives the remainder from \ dividing p by q, treated as polynomials in x."\) In[3]:= PolynomialRemainder[1 + x^3 + x^4 + x^5 + x^6 + x^8 + x^11 + x^13, 1 + x + x^3 + x^4 + x^8, x] Out[3]= -x^7 - x^6 + x^5 + x^4 + x^2 + x + 1 If you want your coefficients to be elements of a finite field, like the integers mod 2, you can use the AbstractAlgebra packages at http://www.central.edu/eaam.html In[9]:= Needs["AbstractAlgebra`Master`"] In[17]:= p = Poly[ZR[2], 1 + x^3 + x^4 + x^5 + x^6 + x^8 + x^11 + x^13]; In[18]:= q = Poly[ZR[2], 1 + x + x^3 + x^4 + x^8]; In[19]:= R = PolynomialsOver[ZR[2]] Out[19]= \!\(TraditionalForm\`"-Ring of Polynomials over Z[2]-"\) In[20]:= PolynomialRemainder[R, p, q] Out[20]= x^7 + x^6 + x^5 + x^4 + x^2 + x + 1 This could be done without the AbstractAlgebra packages using PolynomialMod, but the packages contain a lot of other things. Ken Levasseur Math Sciences UMass Lowell Flip at safebunch.com wrote: > Hello, > > Is Mathematica capable of calculating this type of problem? > > Mod[1 + x^3 + x^4 + x^5 + x^6 + x^8 + x^11 + x^13, > 1 + x + x^3 + x^4 + x^8] > > The second polynomial is irreducible? > > By the way, the soultion is: x^7 + x^6 + 1. > > Thank you for any inputs ...

**References**:**Polynomial Reduction with Mod***From:*Flip@safebunch.com