Generating lots of Examples in Polynomials Rings

Question

I'm studying polynomial rings and i would like to know some tricks for generating lots of examples.

For instance, suppose i'm interested in polynomials over the integers mod (2,x^3 + 1). To get a feel for the structure i would like to expand a bunch of random polynomials and see what happens.

I would like to do this without typing each individually.

Any suggestions?

Thank you!

Answer 1

This hopefully will get you started. It asks for the polynomial order, generate random integers from 1 to 10 for the coefficients of the polynomial, then uses FromDigits to build the polynomial in x. It also asks for the Mod integer value to use. It then calls PolynomialMod and shows the original polynomial and the polynomial mod the integer.

Not pretty, just a quick hack. Just to first see if this is what you want before going more.

Manipulate[
 doit;

 Module[{p, m},
  p = makePolynomial[n, x];
  m = PolynomialMod[p, mod];

  Grid[{
    {"polynomial", Expand[p]},
    {Row[{"polynomial mod ", mod}], m}
    }, Alignment -> Left, Frame -> All]
  ],

 {{n, 2, "polynomial order"}, 2, 10, 1, Appearance -> "Labeled"},
 {{mod, 2, "mod"}, 1, 10, 1, Appearance -> "Labeled"},
 Button["make another", doit = Date[], ImageSize -> 100],
 ContentSize -> {550, 80},

 Initialization :>
  (
   makePolynomial[n_Integer, x_Symbol] := Module[{z, c},
     z = RandomChoice[{-1, 1}] RandomInteger[{1, 10}];
     c = Table[RandomInteger[{-10, 10}], {n}];
     FromDigits[Reverse[AppendTo[c, z]], x]
     ]
   )
 ]