Practice Exam 2
Modern Algebra I, Dave Bayer, April 1, 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 provi
Homomorphisms
1
Definition and examples
Recall that, if G and H are groups, an isomorphism f : G H is a bijection
f : G H such that, for all g1 , g2 G,
f (g1 g2 ) = f (g1 )f (g2 ).
There are many situ
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3. Domain, field, prime and maximal ideal, field of quotients
Definition
(a) A domain is a ring R 6= 0 in which a 6= 0 and b 6= 0 ab 6= 0.
(b) A field is a ring R 6= 0 in which each nonzero element is
1. Ring, unit group, endomorphism ring, polynomial ring
Recall (monoid, group)
A monoid is a set M with an associative multiplication M M M and
a (unique) identity element 1 : Thus
ab.c = a.bc and 1.a
4. Vector space, independent, span, basis, dimension
Recall the two equivalent definitions of an action of a group on a set:
Definition
An action of a group G is a set X, together with
(a) a map G X X
2. Subring, ideal, ring-homomorphism, factor ring
In this and all later sections, ring means commutative ring.
Definition A subset N of
(a) a, b N ab N ;
(b) N contains the identity
Thus a submonoid N
4. Vector space, independent, span (verb and noun), basis
Recall the two equivalent denitions of an action of a group on a set:
DEFINITION
An action of a group G is a set X, together with
(a) a map G
14. Primitive element, embedding formulas, splitting eld
THEOREM A
For each nite extension of elds E D F with F nite or of characteristic 0,
E = F (oz) for some primitive element a E E. (1)
Proof.
F n
Modern Algebra II, Spring 2016
Patrick Gallagher pxgmath.columbia.edu
This course begins with rings, which are simultaneously additive abelian
groups and multiplicative monoids, with addition and mult
7. Polynomial rings, quotient and remainder, gcd in BM
DEFINITION For each ring R, we denote by R[m] the set of all polynomials
oo
f(x) = Z akin, with all ak E R and ak = 0 for suiciently large k. (1)
17. Polynomials in (C[z], FTA, Quadratics, Cubics and Quartics
In this section, all polynomials are in C[:I:], written C[z], and are thought of
as is a complex-valued functions f of a complex variable
ANSWERS TO SOME OF THE HOMEWORK PROBLEMS
Note: these answers are not proofread. Also, the style is somewhat more
terse than I would want to see on a problem set or exam, but I hope this
will give you
Some group tables and group computations
The two groups of order 4 (up to isomorphism): (i) Z/4Z:
0
0
1
2
3
+
0
1
2
3
1
1
2
3
0
2
2
3
0
1
3
3
0
1
2
Aside from the trivial subgroup, Z/4Z has one proper
Practice Exam 2
Modern Algebra I, Dave Bayer, March 24, 2009
Name:
Please work only one problem per page, starting with the pages provided. Clearly label your answer. If a
problem continues on a new p
Exam 2
Modern Algebra 1, Dave Bayer, April 1, 2008
Name: g
[ll (6 pts) [2] (6 pts) [3] (6 pts) {4] (6 pts) [5] (6 pts)
Please work only one problem per page, starting with the pages provided. C
Practice Exam 1
Modern Algebra I, Dave Bayer, February 10, 2009
Each of these problems are model questions with many possible variants, each based on groups of small
order. On the actual exam, specic
Exam 2
Modern Algebra I, Dave Bayer, April 1, 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. Clea
Exam 1
Modern Algebra I, Dave Bayer, February 19, 2008
Name:
[1] (5 pts) [2] (5 pts) [3] (5 pts) [4] (5 pts) [5] (5 pts) [6] (5 pts)
TOTAL
Please work only one problem per page, starting with the page
Exam 3 (Take Home)
Modern Algebra I, Dave Bayer, April 26, 2008
Name:
[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
This take home exam is out o
Exam 1
Modern Algebra I, Dave Bayer, February 17, 2009
Name:
[1] (5 pts) [2] (5 pts) [3] (5 pts) [4] (5 pts) [5] (5 pts) [6] (5 pts)
TOTAL
Please work only one problem per page, starting with the page
Makeup Exam 1
Modern Algebra I, Dave Bayer, February 17, 2009
Name:
[1] (5 pts) [2] (5 pts) [3] (5 pts) [4] (5 pts) [5] (5 pts) [6] (5 pts)
TOTAL
Please work only one problem per page, starting with t
MODERN ALGEBRA I SPRING 2016:
SIXTH PROBLEM SET
1.
An invertible
2 2 matrix A M2 (R) is upper triangular if A =
a b
for some a, b, d R, necessarily with a, d 6= 0. Let B be the
0 d
set of all upper t
10. Roots of a polynomial, minimal polynomial, degree
DEFINITION For each commutative ring R, and each f (x) E R[a:], i.e.
f(a:) = a0 + alzc + + anus", with ac, .,an e R. (1)
and for a E R, we put
f(a