Polymerizable polynomials

[Guzik]

Hello, I have a problem with 2 tasks.

Task 1:
Provide all irreplaceable polynomials grade 2 and 3 in \(\displaystyle \mathbb{Z}_{2}[x]\) and all irrelevant grade 2 polynomials over \(\displaystyle \mathbb{Z}_{3}\).

I am not sure what is the difference between the polynomial of a given degree in \(\displaystyle \mathbb{Z}_{2}[x]\), and the polynomial over \(\displaystyle \mathbb{Z}_{2}\).

Task 2:
Show that for the first p number there is a non-decayable polynomial of degree 2 in \(\displaystyle \mathbb{Z}_{p}[x]\)

Thanks for your help.

 

[HallsofIvy]

There are several words here that do not seem to be standard. How does your text define "polymerizable" polynomial? What are the definitions of "irreplaceable" "irrelevant", and "non-decayable" polynomials? (Are these translations from a non-English text?)

For your other questions, notice the difference between "in Z2[x]" and "over Z2". Z2[x] is the set of all polynomial in the variable x with coefficients base 2. Z2 is the set of integers base 2. A polynomial "in Z2[x]" is a polynomial "over Z2" (as long as the variable is called "x").

By "p number" do you mean "prime number"?