MAT 423 Assignments

Highlighted assignments are due soon.

Due Date

 Assignments
9 Dec Do Exercises 3.64-3.67, 3.70, 3.72, 3.73
Also do 3.80-3.84
Hint for 3.81: We already did it in class (I think)
Hint for 3.83: Use \(h=g\circ f\)
4 Dec Do Exercises 3.49, 3.50, 3.53
Also, Read §3.6 as there will likely be a quiz
20 Nov Do Exercises 3.24, 3.27, 3.29, 3.30, 3.39, 3.41, 3.42(c,d), 3.43
13 Nov Do Exercises 3.12, 3.13, 3.25, 3.26
11 Nov Read §§ 3.2, 3.3
6 Nov Do Exercises 3.9, 3.10
30 Oct Do Exercises 2.48, 2.49, 2.63-2.66
Also:
  1. Explain why the only units of \(\mathbb Z\) are \(\pm1\).
  2. Unlike \(\mathbb Z\), only one element of \(\mathbb Q\) is not a unit. Which is it? How do you know that the other elements are units?
  3. The units of \(\mathbb Z[x]\) are \(\pm1\), while the units of \(\mathbb Q[x]\) are, in a similar way, precisely the units of \(\mathbb Q\). That is, polynomials of degree 1 or higher are not units in these settings. Explain why this is the case.
23 Oct Do Exercises 2.22, 2.23, 2.25, 2.37
21 Oct Read §§ 2.4 and 2.5
16 Oct Do Exercises 2.10, 2.11, 2.12
9 Oct Do Exercises 1.98, 1.100, 1.102, 1.103, 1.104
2 Oct Do Exercises 1.70, 1.71, 1.76, 1.105
Hint for 1.76: Let \(n\in\mathbb N^+\), and let \(m_1=n\). While \(m_i\) is composite, you can find a factorization \(m_i=a_im_{i+1}\) such that \(a_i,m_{i+1}\neq1\). By Lemma 1.44, \(a_i,m_{i+1}\leq m_i\), but neither is 1, so in fact \(a_i,m_{i+1} < m_i\). You have a nonincreasing sequence of integers. Can it go on decreasing indefinitely? If it not, what is the only way it can stabilize? What sort of number do you end up with?
25 Sep Do Exercises 1.52, 1.53(b), 1.55, 1.72, 1.73, 1.75
18 Sep Do Exercises 1.32, 1.33, 1.35, 1.38
11 Sep Read §1.2
Do Exercises 1.17, 1.18, 1.19, 1.31(a,c,e)
4 Sep 1.14, 1.16