Math 5032 - Homework 1
Due 1/27/06
1. Give an example of an R-module M having R-isomorphic submodules
N1 and N2 such that M/N1 and M/N2 are not isomorphic.
2. Suppose V is a nite-dimensional vector space over a eld k, viewed
as a k-module. Describe a comp
Math 5032 - Homework 5
Due 3/03/06
1. If R is a commutative ring with 1 and x1 , x2 are distinct indeterminates
show that R[x1 , x2 ] and R[x1 ] R[x2 ] are isomorphic as R-algebras.
2. If k is a eld and K is an extension eld of k show that Mn (K)
=
K k M
Math 5032 - Homework 7
Due 3/27/06
1. Nakayamas lemma. let R be any ring and M a nitely generated
unitary R-module. Let N = J(R) be the radical of R. If N M = M
show that M = 0.
(Hint: Write M = R x1 , . . . , xs and xs = a1 x1 + + as xs , for ai N .
Obse
Math 5032 - Homework 8
Due 4/05/06
1. Show that a commutative ring R is primitive if and only if R is a eld.
2. Suppose F is an algebraically closed eld and D is a division ring that is
algebraic over F , with F contained in the center of D. Show that D =
Math 5032 - Homework 9
Due 4/17/06
1. Given any two representations T and S of a nite group G on nite dimensional complex vector spaces, show that the inner product
(T , S ) :=
1
|G|
T ( )S ( )
G
is always a real number.
2. Denote by CS the vector space
Math 5032 - Homework 10
Due 4/24/06
1. Suppose a nite group G acts on a nite set S by permutations and let
T denote the permuation representation of G. (H.W. 9, problem 2.)
Let be the character of T .
(a) If G is transitive on S , show that T = 1 L, where
El Saving screenshot.
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dependent variable
the output values in a function (usually y)
it 49
determinant
in a square matrix it is a real number
describing the matrix determined by ad
bc in a 2x2 matrix
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discriminant
the value of the expression
root
a solution of an equation
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zero
a value that gives you 0 for a function
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x-intercept
where the graph crosses the x-axis
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y-intercept
where the graph crosses the y-axis
sir-1
extraneous
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factor (n)
one of two or more numbers or
expressions that when multiplied
together produce a given result
:2: a
factor (v)
rewriting an expression as the product
of its factors
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funcon
a relation in which each element i
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integer
one of the positive or negative whole
numbers or zero
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leading coefficient
the coefcient of the term of highest
degree in a polynomial
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linear function
a function whose graph is a line where
the dependent vari
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end behavior
the directions of a graph as you move
away from the origin
rib
equaon
a statement that two algebraic
expressions are equal
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evaluate
determining the arithmetic value of an
expression by substituting number
valu
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complex number
the real numbers AND the imaginary
numbers
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conjugate
number pairs where the sign of the
radicals or the imaginary numbers are
opposites
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constant
a quantity whose value does not change
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consta
Math 5032 - Homework 6
Due 3/10/06
In this short assignment you will prove the following simple but fundamental fact about group algebras due to Maschke.
Theorem 0.1 Let G be a nite group, F a eld and R = F G the group
algebra of G. Let M be a unitary R-m
Math 5032 - Homework 3
Due 2/17/06
1. Suppose R is a ring with 1. A unitary R-module P is called projecg
tive if given an exact sequence M N 0 of R-modules and an Rhomomorphism f : P N , then there is an R-homomorphism h : P M
such that f = g h.
(a) Show
Math 5032 - Homework 4
Due 2/24/06
Find the characteristic polynomial, invariant factors, elementary divisors,
rational canonical form, and Jordan canonical form (when possible) over Q
for each of the following matrices:
1.
0 4
1 4
2.
c+6
9
4 c6
,cQ
3 2 4
Math 5032 - Homework 2
Due 2/10/06
1. If f : M M is an R-module homomorphism such that f f = f ,
show that M = Kerf Imf.
2. If f : M N and g : N M are R-module homomorphisms such
that g f = 1M (the identity homomorphism of M ), show that N =
Kerg Imf .
3.
axis of symmetry
the line that divides a gure into two
parts that are mirror images
sir 49
binomial
a polynomial of two terms
coefcient
the numerical factor in a term
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complex conjugates
number pairs of the form a + bi and a bi
air 49
complex number
t