How do I find a polynomial in a field?

Question

If I have a polynomial:

$$f(x) = c_0 x^0 + c_1 x^1 + c_2 x^2 + \dots + c_n x^n$$

How can I find the polynomial, modulo a prime number $p$? In other words, I want to take all of the coefficients modulo $p$, and all of the powers modulo $p-1$. So, for instance, the result modulo 3 would be:

$$(c_0 \bmod 3)x^0+$$ $$(c_1 \bmod 3)x^1+$$ $$(c_2 \bmod 3)x^0+$$ $$(c_3 \bmod 3)x^1+$$ $$(c_4 \bmod 3)x^0+$$ $$(c_5 \bmod 3)x^1+$$ $$\dots$$

Please note that this is just an example, and in general I want to find any polynomial modulo any prime $p$.

So, in general, I have something like:

 5 + 13x + 14x^2 + 7x^3 + 8x^4

I want to take all powers of $x$ modulo $p-1$, so I get:

 5 + 13x + 14 + 7x + 8

...which, in turn, equals, by simplifying:

 27 + 20x

Then, taking all coefficients modulo 3, we get:

 0 + 2x

How can I code this?

Answer 1

Given an equation

    eqn = 5 + 13 x + 14 x^2 + 7 x^3 + 8 x^4

We can simply apply a rule to all integers, replacing them the new Mod value. As follows:

    eqn /. x_ /; IntegerQ[x] :> Mod[x, 3]

Which gives:

    3 + 3 x + 2 x^2

The example you give in your question doesn't seem to tally with what you initially describe as your goal?

If you only want to apply the Mod to the exponents, then:

   eqn /. Power[x_, y_] :> Power[x, Mod[y, 3]]

Which gives:

    12 + 21 x + 14 x^2

Answer 2

Should be something like this:

poly = 5 + 13 x + 14 x^2 + 7 x^3 + 8 x^4;
p = 3;
Total@MapIndexed[# x^Mod[First@#2 - 1, p - 1] &, 
  Mod[#, p] & /@ CoefficientList[poly, x]]

6 + 2 x

(I don't follow your example though.. maybe I missed something )