Math 361, Spring 2016, Assignment 4

Carefully define the following terms, then give one example and one non-example of each:

1. Divisibility relation (i.e. $a|b$ where $a$ and $b$ are elements of a domain $D$).
2. Associate (of an element of a domain).
3. Irreducible element (of a domain).
4. Ideal (in any ring $R$).
5. Principal ideal (generated by an element $a$ of a commutative ring $R$).
6. Principal ideal domain (or PID).
7. Standard representation (of an element of the quotient ring $F[x]/\left\langle p\right\rangle$).

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning ideals of $F[x]$.
2. Theorem concerning uniqueness of standard representations in $F[x]/\left\langle p\right\rangle$.

Carefully describe the following algorithms:

1. Ideal membership test for $F[x]$ (i.e. given an arbitrary ideal $I$ in $F[x]$ and an arbitrary polynomial $f\in F[x]$, describe an algorithm that decides whether $f$ is a member of $I$).

Solve the following problems:

1. Prove that the quotient ring $\mathbb{Z}_2[x]/\left\langle x^2+x+1\right\rangle$ is finite by actually counting its elements.
2. Make addition and multiplication tables for the ring defined above. Is this ring a field?
3. Make addition and multiplication tables for $\mathbb{Z}_2[x]/\left\langle x^2+x\right\rangle$. Is this ring a field?
4. Is every finite field isomorphic to $\mathbb{Z}_p$ for some prime number $p$?

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