Math 103B
Final Exam (100 points)
Friday 6/13/2008
Please put your name and ID number on your blue book.
CLOSED BOOK, but BOTH SIDES of two pages of notes are allowed.
Calculators are NOT allowed.
In a multipart problem, you can do later parts without doi
Exam 1 Preparation
The first exam will take place during class on Friday, April 28. There will be assigned seats, which will
be posted on TritonEd. The exam will be closed book and closed notes, and will consist of problems taken
from the list below and 1
Kimberly Chau
MATH 103B, Section A01
April 19,2017
Section 21, Problem 6 Prove Part 2 of Step 3. You may assume any
preceding part of Step 3.
Solution
We start with [(a,b)] + ([(c,d)] + [(e,f)]). Given that [(c,d)] and [(e,f)] are
in F , we know by defini
Laura Jara
MATH 103B, Section A01
April 21, 2017
Section 21, Problem 9 Prove Part 5 of Step 3. You ma assume any
preceding part of Step 3.
Part 5 of Step 3 states that: Multiplication in F is associative.
[(a, b)]([(c, d)][(e, f )]) = [(a, b)][(ce, df )]
Brent Lee
A12054921 MATH 103B (Anne Carter Section A03: 6:00-6:50pm): Homework #2
Problem 20.27
Question: Show that 1 and p 1 are the only elements of the field Zp that are their own multiplicative
inverse. [Hint: Consider the equation: x2 1 = 0]
Solution
f d osuYxw&x6ow ss rw ur t t j w y gw wr j p itr gw v f r ywrt xxQxYtYw bs yuYqsYuYGw D~osuYw xsisxuVs1xyusp ~x d V6z3 i v t p i tr q j t rw ur t j q u w r rwj p tj u ww p w r t tr p rwjr jw q w r q wrw TzT &w 4o)i)Vszut eY63xyuo d osuYxw& rw ur t t j w
q l l l l l l YUy D d qq t h qq t y Yg0cfw_w qq t y s p Y8UY xdp qq t y sy p Y8g0cfw_ xdp 0yrYwsF wpAivyuvwC%cfw_v Y8U0Fs 6YUxdp wt h r qq t y d sqq t y p qq t y d sqq t p Y8U0Fs 6Y8Uy xdp
l d t y U0 h q l d t g0y l ywr tw h i yrw d t y s s d t y p vuYvsF
24. By the definition, an integral domain is a commutative ring with unity 1 0 containing no ze
ro divisors.
Firstly, Since we have proved that the intersection of subrings of a ring is a ring, we can conclude
that an intersection of subdomains of an inte
Math 103B
Partial Solutions
Friday 6/13/2008
1. (a) Suppose ap 6= 0 and (ap)(bp) = 0. Then pq | (ap)(bp). Since gcd(p, q) = 1, q | ab.
Since q is prime, q | b and so bp = 0.
(b) It suffices to show that there is a unity. We need a such that (ap)(bp) = bp
MATH 103B Homework 8 - Solutions
Due June 7, 2013
Version June 8, 2013
Assigned reading: Chapters 18, 20 of Gallian.
Recommended practice questions: Chapter 20 of Gallian, exercises
3, 5, 8, 9, 10, 21
Assigned questions to hand in:
(1) (Gallian Chapter 20
MATH 103B Homework 7
Due May 31, 2013
Version May 31, 2013
Assigned reading: Chapters 18, 19, 20 of Gallian.
Assigned questions to hand in:
(1) (Gallian Chapter 18 #2) In an integral domain, show that a and b are associates if and
only if a b.
Let D be an
MATH 103B Homework 5
Due May 10, 2013
Version May 13, 2013
Assigned reading: Chapters 16-17 of Gallian.
Recommended practice questions: Chapter 16 of Gallian, exercises
17, 21, 26, 27, 43
Chapter 17 of Gallian, exercises
3, 5, 13, 15, 29, 32
Assigned ques
MATH 103B Homework 6 - Solutions
Due May 17, 2013
Version May 18, 2013
Assigned reading: Chapter 17 of Gallian.
Recommended practice questions: Chapter 17 of Gallian, exercises
27, 28, 35
Supplementary Exercises for Chapters 15-18
11, 13, 19, 25
Assigned
MATH 103B Homework 1 - Solutions
Due April 5, 2013
Version April 3, 2013
Assigned reading: Chapter 12 of Gallian.
Recommended practice questions: Chapter 12 of Gallian, exercises
1, 2, 4, 7, 15, 23, 24
Assigned questions to hand in:
(1) (Gallian Chapter 1
MATH 103B Homework 4 - Solutions
Due May 3, 2013
(1) (Gallian Chapter 15 # 2) Prove Theorem 15.2: Let be a ring homomorphism from a
ring R to a ring S. Then Ker, dened as r R : r 0 is an ideal of R.
Solution: We will prove that Ker passes the ideal test:
MATH 103B Homework 3
Due April 19, 2013
Version April 15, 2013
Recommended practice questions: Chapter 13 of Gallian, exercises
35, 45, 47, 49, 51, 62, 63
Chapter 14 of Gallian, exercises
8, 9, 12, 13, 17, 24, 28, 29
Assigned questions to hand in:
(1) (Ga
MATH 103B Homework 2 - Solutions
Due April 12, 2013
Version April 12, 2013
Assigned reading: Chapters 12-13 of Gallian.
Recommended practice questions: Chapter 12 of Gallian, exercises
30, 31, 32, 33
Chapter 13 of Gallian, exercises
1, 2, 5, 6, 9, 15, 18
h g q p us o x h s x ps x w h p u x x x x x qtTYY|ptt`yhuTd &yltY3uq1qd~vt6tqT `tYYpuddTvqidtDtYidtDuvu1jVYt v QbyxV3qx q p uso p sx q x q p uso x ls oxs x xs ux h g g h q q puso x h s x ps ryt~tTYY|pttQyhYu|t x w x x x x h p u g q p us o p sx q x s x tus
z
z
z
| e 7z g z z z z Xz ~ z 9) k ~ eq | | g | | E~ ~ 7z z g z z #XD43lSteXtIYu | v v s e r p x sr e | k ~ | e e ~ X g 7z g X| ~ g z qeS g X| X ~ g 7z f | ~ e gn s n v k sr p e ymwvyUyn | aeeUequ gn n s k s n v k sr p e ymwvyUyGwtwSlaeeUeGf | z z
f xu i rxt x t u u x u q x i xt xu w x u Tq6sg6q`3gsVgs1g`vQ6gkD| g u i i i tx qtu u r iutx u x ix w x u u tx w i x 6yxgqrdqTyyugs| g cfw_y`g3T`cfw_yxdsgqqg`g3cfw_QTVkTG f qt uu | x w i u q t u p a`yuyuTdsD6vg1&gk`psq l | l l p l p q p ` 33vclgG p clpsgq
Applied Algebra Lecture 1
Audrey Terras
Math. Dept., U.C.S.D., San Diego, Ca 92093-0112 email: [email protected] web: http:/math.ucsd.edu/~aterras/ March 28, 2010
The Integers
The set of integers is Z = f0; 1; 2; 3; : : :g : It is dened by various axioms.
Math 103B
Final Exam Solutions
23 March 2006
1. Since it is a splitting eld, the extension is Galois, so we can use the fundamental theorem of Galois theory for (a)(c). (a) The degree of the extension is the order of the Galois group, which is 8.
Math 103B
First Hour Exam (40 points)
1 February 2006
Please put your name and ID number on your blue book. The exam is CLOSED BOOK except for one page of notes. Calculators are NOT allowed. In a multipart problem, you can do later parts wit
Math 103B
First Hour Exam Solutions
1 February 2006
1. (a) Since x2 = 1 in R, there are just four elements: A, 1 + A, x + A and x + 1 + A, where A = x2 + 1 >. (b) We can exhibit zero divisors: (x + 1 + A)2 = x2 + 2x + 1 + A = x2 + 1 + A = A. 2. Si
Math 103B
Second Hour Exam (50 points)
8 March 2006
Please put your name and ID number on your blue book. The exam is CLOSED BOOK except for two pages of notes. Calculators are NOT allowed. In a multipart problem, you can do later parts with