Math 361, Spring 2019, Assignment 7

Read:

1. Section 23.

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

1. $f\%g$ (the remainder when $f$ is divided by $g$).
2. Root (of a polynomial).
3. Ideal.
4. $\left\langle a\right\rangle$ (the principal ideal generated by $a$).

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

1. Theorem concerning polynomial long division.
2. Factor theorem.
3. Theorem concerning factorization of quadratic and cubic polynomials.
4. Eisenstein's criterion.
5. Criterion for a principal ideal to be contained in an arbitrary ideal.
6. Criterion for a principal ideal to be contained in another principal ideal.
7. Criterion for two principal ideals to be equal.

Carefully describe the following algorithms:

1. Sieve of Eratosthenes.

Solve the following problems:

1. Section 23, problems 1, 3, 9, 11, 12, 13, 14, 18, 19, 20, 21, and 29.

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