Math 361, Spring 2020, Assignment 8

Read:

1. Section 23.

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

1. $f\%g$ (the remainder or residue when $f$ is divided by the non-zero polynomial $g$).
2. Standard representative (of a coset $f+\left\langle m\right\rangle$ in the quotient ring $F[x]/\left\langle m\right\rangle$).
3. $\alpha$ (the standard generator of the ring $F[x]/\left\langle m\right\rangle$).
4. Divisibility relation (on a domain $D$).
5. Associate relation (on a domain $D$).
6. Associate class (of an element $a\in D$).

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

1. Theorem concerning polynomial long division.
2. Theorem relating $f+\left\langle m\right\rangle$ to $(f\%m)+\left\langle m\right\rangle$.
3. Equality test for elements of $F[x]/\left\langle m\right\rangle$.
4. Non-redundant list of the elements of $F[x]/\left\langle m\right\rangle$ (i.e. each element of this ring can be written uniquely in the form $f+\left\langle m\right\rangle$ where $\deg(f)$ is...)
5. Theorem concerning $m(\alpha)$ (the "basic principle of arithmetic in $F[x]/\left\langle m\right\rangle$").
6. Theorem concerning the basic properties of the divisiblity relation.
7. Theorem concerning the basic properties of the associate relation.
8. Theorem characterizing associate classes in terms of units.
9. Theorem describing associate classes in $\mathbb{Z}$.
10. Theorem describing associate classes in $F[x]$.

Solve the following problems:

1. Section 23, problems 1, 2, 3, 4, and 9. (Hint for 9: first find a root of the givem polynomial, if necessary by brute force. For instance, $4$ is a root, since $4^4+4=2^8+4=256+4=260=0$ (mod $5$). Then, divide the polynomial by $x-4$ and see what happens. You will be left with a degree $3$ quotient $q$. Try to find a factor of $q$ by the same procedure, i.e. by finding a root $a$ of $q$ and then dividing $q$ by $x-a$. Repeat until you have solved the problem.)
2. Consider the ring $R=\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle$. First, make a non-redundant list of the elements of $R$. (Hint: there are exactly eight elements.) Then make addition and multiplication tables. (This is time-consuming, but at least make a credible partial effort.) Finally, show that $R$ is in fact a field. (It is called $GF(8)$ in the literature.)
3. Working in the field $GF(8)$ defined above, compute the multiplicative inverse of $\alpha^2+1$. (Hint: at this stage, it is a table lookup. We will discuss a better inversion method soon.)
4. Suppose $F$ is any field. Describe the associate classes of $F$. (Hint: there are only two classes.)

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