MATH:5000 - Fall 2016
Homework
Assignment 10
Reading (from Dummit and Foote):
Read Sections 8.1 - 8.3, 9.1 - 9.3.
Please look through the sections for EACH lecture before the lecture.
See the Lecture Schedule for the sections that will be covered.
Then after EACH lecture, review your lecture notes and
thoroughly read the sections covered in class.
Part A Problems (to be handed in on
Tuesday, November 15, at the beginning of the discussion section):
- Section 8.1: # 4, 11.
- Section 9.1: # 6, 7.
IMPORTANT: In #7, you should NOT use results from Section 9.2 to argue
(but look at Section 8.2).
Part B Problems (to be handed in on
Friday, November 18, at the beginning of the lecture):
If you hand in ALL Part B Problems together with the Part A Problems already on
Tuesday, November 15, 2 extra points will be added to your homework score.
- Section 8.2: # 4, 6.
Hint for #4: If I ≠ 0 is an ideal of R, show that condition (ii)
implies that the set S = { (x) | x ∈ I } has a maximal element under
inclusion. Then use condition (i) to show that this maximal element is I.
Hint for #6: In part (c) use the definition of J to argue that
s ∈ J.
- Section 8.3: # 5(a)(b).
IMPORTANT: R = { a + b √-n | a, b ∈ Z }.
In part (b), the quadratic integer ring O is equal to R
when D = -n and D ≡ 2, 3 mod 4 (see p. 229). So the conclusion follows immediately once you have shown that R is not a UFD.
Additional Practice Problems (you have
to know how to do these, but you do NOT have to hand these in):
- Section 8.1: # 3, 6.
- Section 8.2: # 1, 2, 3, 8.
- Section 8.3: # 1, 2, 8.
- Section 9.1: # 3, 4, 5, 13, 14, 15, 16, 17, 18.
Frauke Bleher
Nov 08 2016