Name:
Math 236
Spring 2012
Dr. Seelinger
Field Extenstions and Factoring
For this worksheet, we see applications of Theorems 5.10 and 5.11.
1. Consider the polynomial p(x) = x3 + 2x + 1 in Z3 [x]. Prove that it is irreducible in Z3 [x].
By Corollary 4.18,
Name:
Math 236
Spring 2012
Dr. Seelinger
Homework 14, Section 5.3 Solutions
Sect 5.1, Problem 1: Determine whether the given congruence-class ring is a eld.
(a) Z3 [x]/(x3 + 2x2 + x + 1). By Theorem 5.10, this is a eld if and only if p(x) = x3 + 2x2 + x +
NAME:
Math 236
Prof. Seelinger
Spring 2012
REVIEW PROBLEMS
1. Use congruences to prove that 3n2 + 1 is never divisible by 5 for any integer n.
2. Let n Z with n > 1. Let a, b Z be such that (a, n) = 1. Prove that the equation
ax = b has a unique solution
Name:
Math 236
Spring 2012
Dr. Seelinger
EXAM III - Solutions
Answer all of the following problems. Each problem is worth 20 points.
Fully Justify Each Answer.
1. Prove that the following:
(a) x5 + x4 + x2 + x + 1 is irreducible in Z2 [x].
Let f (x) = x5
Name:
Math 236
Spring 2012
Dr. Seelinger
EXAM II - Solutions
Answer all of the following problems. Each problem is worth 20 points.
Fully Justify Each Answer.
1. Let R be the set of real numbers with regular addition and with multiplication dened by a
a2
Name:
Math 236
Spring 2012
Dr. Seelinger
EXAM I - Solutions
Answer all of the following problems. Each problem is worth 20 points.
Fully Justify Each Answer.
1. Let a, b Z so that not both are zero. Prove that if (a + b, a b) = 1 then (a, b) = 1. Is the
c