Re: Finding Roots of Polynomials over Polynomial Quotient Rings

• To: mathgroup at smc.vnet.net
• Subject: [mg93502] Re: Finding Roots of Polynomials over Polynomial Quotient Rings
• From: Julian <julianhaw at gmail.com>
• Date: Wed, 12 Nov 2008 06:46:02 -0500 (EST)
• References: <200811091025.FAA20508@smc.vnet.net> <gf8ree\$pmh\$1@smc.vnet.net>

```On Nov 10, 3:30 am, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> On 9 Nov 2008, at 19:25, Julian wrote:
>
> > Hi all,
>
> > Just wondering how I would go about finding roots of an polynomial,
> > say t^2 +2, over a quotient ring, say Z/5Z[x] / <x^2+2>, in
> > Mathematica?  The modulus option in Reduce is only for integers, and
> > I've had trouble finding anything relevant in the finite fields
> > package... anyone know if this can be done in Mathematica?
>
> > Thanks,
> > Julian
>
> Since a finite field has only finitely many elements and since the
> FiniteFields package lets you perform arithmetic on these elements,
> you can always find the roots of a polynomial simply by substituting
> all the field elements into the polynomial and selecting those that
> satisfy it. Or am I misunderstanding your question?
>
> Andrzej Kozlowski

Yep it worked!  Thanks.  I made a function to list all the elements in
the field, then simply substituted each one

```

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