Math 361, Spring 2019, Assignment 8
From cartan.math.umb.edu
Read:
- Section 26.
Carefully define the following terms, then give one example and one non-example of each:
- $R/I$ (the quotient of the ring $R$ by the ideal $I$).
- Product (of two cosets in $R/I$).
- Standard representative (of a coset in $F[x]/\left\langle m\right\rangle$).
- $\mathbb{C}$.
- $GF(4)$.
Carefully state the following theorems (you need not prove them):
- Theorem relating kernels to ideals.
- Theorem concerning unique representation of elements of $F[x]/\left\langle m\right\rangle$.
- Procedure to calculate the standard representative of a coset in $F[x]/\left\langle m\right\rangle$.
Solve the following problems:
- Section 26, problems 11, 12, 13, 14, and 15.
- 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.)
- ($\mathbb{Q}[\sqrt[3]{2}]$) 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.