Math 361, Spring 2017, Assignment 5

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

1. Ring homomorphism.
2. Unital (ring homomorphsim).
3. Image (of a ring homomorphism).
4. Kernel (of a ring homomorphism).
5. Ideal (in a ring $R$).
6. Principal ideal (generated by an element $a$ of $R$).
7. Quotient ring (i.e. $R/I$ where $R$ is a ring and $I$ is an ideal).

Carefully describe the following algorithms:

1. Sieve of Eratosthenes (for generating lists of prime numbers).
2. Sieve of Eratosthenes (for generating lists of irreducible polynomials over finite fields).
3. Factorization algorithm for polynomials with coefficients in finite fields.

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

1. Theorem relating factorization of quadratic and cubic polynomials to roots.
2. Theorem relating factorization in $\mathbb{Q}[x]$ to factorization in $\mathbb{Z}[x]$.
3. Eisenstein's criterion.
4. Theorem concerning images and pre-images of subrings.
5. Theorem relating kernels to ideals.

Solve the following problems:

1. Section 23, problems 14, 15, 19, 21, and 29.
2. Section 26, problems 3, 4, 11, 12, 13, and 14.
3. Find all irreducible quadratic polynomials with coefficients in $\mathbb{Z}_3$.
4. Working in $\mathbb{Z}_2[x]$, factor the polynomial $f = x^5+x^4+1$ into irreducibles. (Hint: you do not need to run the Sieve all the way to degree four. In this problem it is enough to run it to degree two, then rely on other principles.)

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