Math 380, Spring 2018, Assignment 1

The beginner ... should not be discouraged if ... he finds that he does not have the prerequisites for reading the prerequisites.

- P. Halmos

Read:

1. Section 1.1.

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

1. Field.
2. $n$-dimensional affine space over a field $\mathsf{k}$.
3. Multi-index.
4. Monomial $x^\alpha$ associated with the multi-index $\alpha$.
5. Polynomial.
6. $\mathsf{k}[x_1,\dots,x_n]$.
7. Sum (of two polynomials).
8. Product (of two polynomials).
9. Degree (of a monomial).
10. Degree (of a polynomial).
11. Value (of a polynomial $p\in\mathsf{k}[x_1,\dots,x_n]$ at a point $(a_1,\dots,a_n)\in\mathsf{k}^n$).
12. Function induced by a polynomial.

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

1. Theorem on polynomial long division.
2. Factor theorem.
3. Theorem on the degree of a product.

Solve the following problems:

1. Section 1.1, problems 1, 2, and 5.
2. Consider the polynomial $f=x^3-8$ in $\mathbb{R}[x]$. What is the remainder when $f$ is divided by $x-2$? (First try to answer this question without actually doing the long division. Then verify your answer by long division.)
3. (Bound on the number of roots of a univariate polynomial) Suppose $\mathsf{k}$ is any field, and suppose $f\in\mathsf{k}[x]$ is a univariate polynomial of degree $d\geq0$. Prove that $f$ has at most $d$ roots. (Hint: combine the factor theorem with the formula for the degree of a product.)
4. What is the degree of the polynomial $x^3 + x^2yz$?

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