Math 361, Spring 2018, Assignment 3

Read:

1. Section 22.

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

1. Polynomial (with coefficients in the commutative unital ring $R$).
2. $R[x]$.
3. Degree (of a polynomial).
4. Canonical injection (of $R$ into $R[x]$).
5. Constant polynomial.

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

1. Euler's Theorem.
2. Fermat's Little Theorem.
3. Bound on the degree of a sum.
4. Bound on the degree of a product.
5. Exact formula for the degree of a product, when coefficients come from an integral domain.
6. Theorem characterizing when $R[x]$ will be an integral domain.
7. Theorem characterizing when $D[x]$ will have unique factorization.
8. Theorem characterizing the units of $D[x]$.

Carefully describe the following algorithms:

1. Long division of polynomials (in $F[x]$, where $F$ is any field).

Solve the following problems:

1. Section 20, problems 4, 5, and 10.
2. Section 22, problems 1, 3, 5, 12, 13, and 14.
3. Working in $\mathbb{Q}[x]$, find the quotient and remainder when $x^2-x-6$ is divided by $x-2$.
4. Working in $\mathbb{Q}[x]$, find the quotient and remainder when $x^2-x-6$ is divided by $x-3$.
5. Working in $\mathbb{Z}_7[x]$, find the quotient and remainder when $x^3+2x+2$ is divided by $x-2$.
6. Working in $\mathbb{Z}_7[x]$, factor the polynomial $x^3+2x+2$ as a product of linear factors. (Hint: your result in problem 13 is helpful here.)

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