Algebraic Structures, Fall 2014
Homework 4 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 2.7.2). Let G be any group, g a xed element of G. Dene : G G by (x) = gxg 1 . Show that is a bijective homomorphism
from G to
Algebraic Structures, Fall 2014
Homework 3 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 2.5.1). If H and K are subgroups of G, show that
H K is a subgroup of G.
Lets go ahead and prove the more general fact: if Hi
Algebraic Structures, Fall 2014
Homework 2 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 2.3.21). Let G be the set of all real 2 2 matrices
a b
, where ad = 0. Prove that G forms a group under matrix multipli0 d
ca
Algebraic Structures, Fall 2014
Homework 1 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 1.2.8). If the set S has a nite number of elements,
prove the following:
(a) If maps S onto S, then is one-to-one.
(b) If is
Algebraic Structures, Fall 2014
Homework 5 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 2.10.10). Determine which of the following are even
permutations:
(a) (1, 2, 3)(1, 2)
(b) (1, 2, 3, 4, 5)(1, 2, 3)(4, 5)
(c)
Algebraic Structures, Fall 2014
Homework 6 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 2.11.3). List all the conjugate classes in the group of
quaternion units (see Problem 21, Section 2.10), nd the ca s and veri
Algebraic Structures, Fall 2014
Homework 8 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 2.13.1). If A and B are groups, prove that A B is
isomorphic to B A.
Dene the obvious ippy map : A B B A by (a, b) = (b, a).
Algebraic Structures, Fall 2014
Homework 11 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 3.Supp.1). Let R be a commutative ring; an ideal P
of R is said to be a prime ideal of R if ab P , a, b R implies that a P
o
Algebraic Structures, Fall 2014
Homework 12 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 3.8.1). Find all the units in Z[i].
Note: I again change notation from the book to use Z[i] rather than J[i]
to denote the r
Algebraic Structures, Fall 2014
Homework 13 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 3.10.1*). Let D be a PID, F its eld of fractions.
Prove the Gauss Lemma for polynomials with coecients in D factored as
prod
Algebraic Structures, Fall 2014
Homework 14 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 5.3.5). Putting = (1 + i 3)/2, prove that Q() is
the splitting eld of x4 + x2 + 1 over Q.
We rst note that x4 +x2 +1 has no
Algebraic Structures, Fall 2014
Homework 10 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 3.2.4). If every x R satises x2 = x, prove that R
must be commutative.
We rst show that every element is its own additive in
Algebraic Structures, Fall 2014
Homework 9 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 3.2.6*). If D is an integral domain and D is of nite
characteristic, prove that the characteristic of D is a prime number.
Th
Algebraic Structures, Fall 2014
Homework 7 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Herstein 2.12.7). Let G be a group of order 30.
(a) Show that a 3-Sylow subgroup or a 5-Sylow subgroup of G must be normal
in G.
(b) F