Take-Home Exam 2
Modern Algebra II, Dave Bayer, October 28, 2008
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts)
[1] Make a list of small primes for the Gaussian integers Z[i].
TOTAL
[2] Dene a principal ideal domain. State the ascending
Exam 11
Final Exam
Modern Algebra II, Dave Bayer, December 22, 2009
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts) [6] (6 pts)
TOTAL
Please work only one problem per sheet of paper; each problem will be graded separately. Clearly label
First Midterm Exam
Modern Algebra, Dave Bayer, February 17, 1999
Name:
ID:
School:
[1] (6 pts)
[2] (6 pts)
[3] (6 pts)
[4] (6 pts)
[5] (6 pts)
TOTAL
Each problem is worth 6 points, for a total of 30 points. Please work only one problem per
page, and label
Final Examination
Dave Bayer, Modern Algebra, May 13, 1998
[1] Which of the following rings are integral domains? Explain your reasoning.
(a) F2 [x]/(x4 + x2 + 1)
(b) Q[x]/(x4 + 2x2 + 2)
[2] Consider the ideal I = (2, 3x2 ) Z[x], and let R = Z[x]/I .
(a)
Final Exam
Modern Algebra, Dave Bayer, May 12, 1999
Name:
ID:
School:
[1] (5 pts)
[2] (5 pts)
[3] (5 pts)
[4] (5 pts)
[5] (5 pts)
[6] (5 pts)
[7] (5 pts)
[8] (5 pts)
TOTAL
Please work only one problem per page, and label all continuations in the spaces
pr
Second Midterm Exam
Modern Algebra, Dave Bayer, March 31, 1999
Name:
ID:
School:
[1] (6 pts)
[2] (6 pts)
[3] (6 pts)
[4] (6 pts)
[5] (6 pts)
TOTAL
Each problem is worth 6 points, for a total of 30 points. Please work only one problem
per page, and label a
Practice problems for rst midterm, Spring 98
midterm to be held Wednesday, February 25, 1998, in class
Dave Bayer, Modern Algebra
All rings are assumed to be commutative with identity, as in our text.
[1] Prove that if a ring R has no ideals other than (0
Practice Problems for Final Exam
Modern Algebra, Dave Bayer, May 3, 1999
Our nal will be held on Wednesday, May 12, 1:10pm 4:00pm, in our regular classroom. It will consist of 8 questions worth 40 points in all. Two questions will be review
from previous
Practice problems for nal exam
Final to be held Wednesday, May 13, 1:10pm - 4:00pm
We will have a problem session in preparation for this nal:
Monday, May 11, 8:00pm - 10:00pm, 507 Mathematics
[1] Let f (x, y, z ) = x2 y + x2 z + xy 2 + y 2 z + xz 2 + yz
Practice problems for rst midterm, Spring 98
midterm to be held Wednesday, February 25, 1998, in class
Dave Bayer, Modern Algebra
All rings are assumed to be commutative with identity, as in our text.
[1] Prove that if a ring R has no ideals other than (0
Practice Problems for Second Midterm Exam
Modern Algebra, Dave Bayer, March 29, 1999
This rst set of problems are collected from various materials already posted on the web.
[1] Working in the Gaussian integers Z[i], factor 2 into primes.
[2] Let a = 3 i
Practice problems for second midterm
midterm to be held Wednesday, April 8, in class
Dave Bayer, Modern Algebra
We will have a problem session in preparation for this midterm:
Monday, April 6, 8:00pm - 10:00pm, 507 Mathematics
[1] Prove the Eisenstein cr
Second midterm
Dave Bayer, Modern Algebra, April 8, 1998
[1] Prove the Eisenstein criterion for irreducibility: Let f (x) = an xn +
. . . + a1 x + a0 Z[x], and let p be a prime. If p doesnt divide an , p does
divide an1 , . . . , a0 , but p2 doesnt divide
First Exam
Modern Algebra II, Dave Bayer, October 5, 2010
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts)
TOTAL
Please work only one problem per page, starting with the pages provided. Clearly label your answer. If a
problem continues on
Practice Final Exam
Modern Algebra II, Dave Bayer, December 2010
Name:
[1] (4 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts) [6] (6 pts) [7] (6 pts)
TOTAL
Please work only one problem per page, starting with the pages provided. Clearly label your an
Take-Home Exam 1
Modern Algebra II, Dave Bayer, September 23, 2008
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts)
TOTAL
Please work only one problem per page, starting with the pages provided. Clearly label your answer. If a
problem con
Practice material for Galois theory I
1. Mark the squares that are followed by correct statements.
There exists a eld with 10 elements.
There exists a eld with 1024 elements.
Any nite extension of Q is a splitting eld of some
polynomial.
Any nite extensio
First Exam
Modern Algebra II, Dave Bayer, October 8, 2009
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts)
TOTAL
Please work only one problem per page, starting with the pages provided. Clearly label your answer. If a
problem continues on
Exam 1
Modern Algebra II, Dave Bayer, October 2, 2008
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts)
TOTAL
Please work only one problem per page, starting with the pages provided. Clearly label your answer. If a
problem continues on a n
Exam 2
Modern Algebra II, Dave Bayer, November 6, 2008
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts)
TOTAL
[1] Dene a primitive polynomial, for polynomials in Z[x]. Prove Gausss lemma: The product of two primitive
polynomials is primit
Exam 2
Modern Algebra II, Dave Bayer, November 19, 2009
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts)
TOTAL
Please work only one problem per page, starting with the pages provided. Clearly label your answer. If a
problem continues on a
Final Exam
Modern Algebra II, Dave Bayer, December 16, 2008
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts)
TOTAL
[1] Prove the Eisenstein Criterion: If f(x) Z[x] and p is a prime, such that the leading coecient of f(x) is
not divisible
Practice Final
Modern Algebra II, Dave Bayer, December 4, 2008
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts)
TOTAL
[1] Prove the Eisenstein Criterion: If f(x) Z[x] and p is a prime, such that the leading coecient of f(x) is
not divisib
Practice Problems
Modern Algebra II, Dave Bayer, September 29, 2009
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts)
TOTAL
Please work only one problem per page, starting with the pages provided. Clearly label your answer. If a
problem co
Additional Practice Problems for Exam 2
Modern Algebra II, Dave Bayer, November 12, 2009
Name:
[1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts)
TOTAL
Please work only one problem per page, starting with the pages provided. Clearly label your an
Modern Algebra II, Quiz 2, Tuesday Oct 25.
NAME:
1 (10 points). Mark the squares that are followed by
correct statements. We assume that M is a left module
over a ring R and M1 , M2 are submodules of M .
The intersection M1 M2 is a submodule of M .
The un
Modern Algebra II, Quiz 1, Tuesday Oct 4.
NAME:
1 (20 points). Mark the squares that are followed by
correct statements.
Any ring is commutative.
Any eld is commutative.
Any integral domain is a principal ideal domain.
Direct product F1 F2 of two elds is
Name:
Modern algebra II. Midterm exam. November 3,
2011.
Textbooks and notebooks cannot be used during the exam.
1. (20 points) Mark those squares that are followed by
correct statements.
The intersection of two ideals of a commutative ring R
is an ideal