Math 361, Spring 2019, Assignment 8

Read:

1. Section 26.

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

1. $R/I$ (the quotient of the ring $R$ by the ideal $I$).
2. Product (of two cosets in $R/I$).
3. Standard representative (of a coset in $F[x]/\left\langle m\right\rangle$).
4. $\mathbb{C}$.
5. $GF(4)$.

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

1. Theorem relating kernels to ideals.
2. Theorem concerning unique representation of elements of $F[x]/\left\langle m\right\rangle$.
3. Procedure to calculate the standard representative of a coset in $F[x]/\left\langle m\right\rangle$.

Solve the following problems:

1. Section 26, problems 11, 12, 13, 14, and 15.
2. Construct a field with exactly nine elements. (Hint: first come up with a specific irreducible quadratic polynomial $m\in\mathbb{Z}_3[x]$. Then make a multiplication table for $\mathbb{Z}_3[x]/\left\langle m\right\rangle$, and see what happens.)
3. ($\mathbb{Q}[\sqrt[3]{2}]$) Show that the polynomial $x^3-2$ is irreducible over $\mathbb{Q}$. Then, describe the elements of the ring $\mathbb{Q}[x]/\left\langle x^3-2\right\rangle$. Finally, list several specific elements, and show how to multiply them.

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