Abstract Algebra

Instructions

• Submit your file by 11:59 pm on the due date. Late submissions will not be accepted.
• If you'd like, you can use the following LaTeX Homework Template.
• In all assignments, give justifications for your answers. For example, if you claim that a certain relation is not an equivalence relation because it is not transitive then show that it is not transitive.
• Problems taken from Chapter 2 - The Integers

Problems

1. Problem 6
2. Problem 15: Only parts (a), (b), (c)
3. Problem 16
4. Problem 22
5. Problem 27
6. BONUS: Problem 19. I found a "solution" online to this problem. However, I think that it is incomplete and, if I were you, I would try another approach. Hint: The number 2 is the only even prime. Thus, if $xy=a^2$ and $a=p_1{p}_2\cdots p_n$ for primes $p_i$ and with $p_1\le p_2\le \cdots \le p_n$, then either $p_1=2$ or $p_1>2$. Treat each case separately. The first case $p_1=2$ is easy. For the second case use induction on the number $n$ of primes. The second case just means that all the primes $p_i$ are odd and every odd integer $p$ is of the form $p=2q+1$. There's probably a shorter proof.