The problem is to confirm that the ideal generated by $x^3+x+1$ in $\mathbb{Z}/2\mathbb{Z}[x]$ is prime and the ideal generated by $x^3-x-1$ is maximal in $\mathbb{Z}/3\mathbb{Z}[x]$.
I tried to take quotient ring and see if they are integral domain or field. But it was hard for me to tell if the resulting quotient rings are integral domain or field. Any help will be greatly appreciated.
On
Since $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$ are fields, their respective polynomial rings are Euclidean domains, hence PIDs. It therefore suffices to check that $x^{3}+x+1$ and $x^{3}-x-1$ are irreducible over each respective polynomial ring. Feel free to comment further if you need more than this.
The second problem: The coefficients have only 3 possibilities, 0,1 or 2. And the power of $x$ in the quotient ring by the ideal of $x^3-x-1$ has to be at most two and so the quotient ring will have $3\times3\times3=27$ elements. Considering you want to avoid any concept like irreducibility you can simply write the multiplication table for all the non-zero elements and realise 1 as a product making everything invertible, hence a field.
For example constants have inverses, and in the quotient ring $x^3-x-1=0$ which is the same as saying $x^3-x=1$ or $x$ and $x^2-1$ are inverses of each other.
And $(x^2-1)^2$ will be the inverse of $x^2$.
First problem is also easier: you have just 4 elements in the quotient ring and need to carry out all multiplications and see that product of non-zero elements yield again a non-zero element.
After doing all these and convincing yourself that those ideals are prime and maximal you will realize that alternatives such as Euclidean domain concepts may be worth learning in order to avoid massive calculations.