Math 361, Spring 2016, Assignment 5

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

Theorem relating maximal ideals to fields.
Theorem relating irreducibility of $p\in F[x]$ to maximality of $\left\langle p\right\rangle$.
Theorem relating $f(a)$ to the remainder when $f$ is divided by $x-a$ (in class I called this the "synthetic division theorem").
Factor theorem.
Theorem concerning irreducibility of quadratic and cubic polynomials.
Rational Root Theorem.
Eisenstein's criterion.

Section 23, problems 9 (hint: first find a root, then use the Factor Theorem and long division, then repeat), 14, 15, 16, 19, and 21.
Section 27, problems 5 and 7.
(Sieve of Eratosthenes for integers) List the integers from 2 to 30, in order. Circle the first integer on the list, then delete all of its multiples. Circle the next integer still on the list, then delete all of its multiples. Continue until all integers on the list have been either circled or deleted. Which integers have been circled?
(Sieve of Eratosthenes for polynomials over a finite field) Make a list consisting of all linear polynomials in $\mathbb{Z}_2[x]$ (in some order), followed by all quadratic polynomials (in some order), then all cubic polynomials, then all quartic polynomials. Circle the first polynomial on the list, then delete all of its multiples. Circle the next polynomial still on the list and delete all of its multiples. Continue until all polynomials on the list have been either circled or deleted. Which polynomials have been circled?
Construct a field with exactly sixteen elements. (You do not need to make addition or multiplication tables, but please do identify your field unambiguously, and work a few sample calculations.)
Can you envision a method that would produce a field with 125 elements? What about a field with 6 elements?

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