Written Homework 10 Solution
MATH 431
Radford
04/14/2009
1. Page 449, number 8: (20 points) Let G = <, >, where = (1 2)(3 4) and = (2 4). Then
G = <, > and b = = (1 2)(3 4)(2 4) = (1 2 3 4). Set a = . Then a2 = b4 = I ; indeed the
order of a is 2 and the
MATH 431
Written Homework 8 Solution
Radford
04/09/2009
np denotes the number of Sylow p-subgroups in a nite group G.
1. Page 413, number 8: (20 points) Since |A4 | = 4!/2 = 12 = 22 3, each Sylow 3subgroup of A4 has 3 elements, and is thus generated by s3
MATH 431
Written Homework 7 Solution
Radford
04/09/2009
n
1. Page 388, number 20: (20 points) g (x) Zp [x] is irreducible and divides xp x
n
n
in Zp [x]. Let F be a splitting eld of xp x over Zp . Then |F | = pn and xp x =
pn
x in Zp [x]
aF (x a); see th
MATH 431
Written Homework 6 Solution
Radford
04/13/2009
1. Page 377, number 6: (20 points) Regard g (x) F [x] as a polynomial with coecients
in F [a] and let b be a zero of g (x) belonging to some eld extension of F [a]. We need
only show that Deg g (x) =
MATH 431
Written Homework 5 Solution
Radford
04/09/2009
1. Page 349, number 30: (20 points) Let w W (5). Then T (v ) = w for some
v V since T is onto (5). Since cfw_v1 , . . . , vn spans V there are a1 , . . . , an F such
that v = a1 v1 + + an vn (5). Si
MATH 431
Written Homework 4 Solution
Radford
02/07/09
Let R be a commutative ring with unity. Recall that R denotes the multiplicative
group of units of R. Let a R. Throughout R = D is an integral domain.
1. Page 334, number 22: (20 points) We base our so
MATH 431
Written Homework 3 Solution
Radford
02/07/09
Let R be a commutative ring with unity. Recall that R denotes the multiplicative
group of units of R. Let a R. We have shown that
<a> = R, that is Ra = R, if and only if a R .
(1)
Throughout R = D is a
MATH 431
Written Homework 2 Solution
Radford
02/05/09
1. Page 315, number 4: (20 points) Write r = p/q , where p, q Z and have no
common prime factor. Since r is a root of f (x) we may write f (x) = (x r)g (x) for
some g (x) Q[x]. Clearing denominators ag
MATH 431
Written Homework 1 Solution
Radford
02/01/09
1. Page 268, number 4: (20 points) Let S = cfw_(n, n) | n Z (7). Then S = as
(0, 0) S . Suppose (m, m), (n, n) S . The calculations
(m, m) (n, n) = (m n, m n) S (4)
and
(m, m)(n, n) = (mn, mn) S (4)
sh
MATH 425
Hour Exam II Solution
Radford
04/19/2009
1. (25 points) Important fact about the minimal polynomial are found in Theorems 20.3, 21.2,
and 21.3. These can be used without explicit reference.
(a) Q( 5+ 3) Q( 5, 3) (2). Since ( 5+ 3)( 5 3) = 5+3 = 8
MATH 425
Hour Exam II
Radford
04/17/2009
Name (print)
(1) Return this exam copy with your exam booklet. (2) Write your solutions in your exam booklet.
(3) Show your work. Answers must be justied. (4) There are four questions on this exam. (5)
Each questio
Hour Exam I Solution
MATH 425
Radford
02/22/2009
For a commutative ring R with unity recall that R denotes the multiplicative group of units of
R. Z denotes the ring of integers, Q and R denote the eld of rational numbers and real numbers
respectively.
1.
MATH 431
Written Homework 12 Solution
Radford
04/20/2009
1. Page 560, number 4: (20 points) set E = Q( 2, 5, 7). Then E is a splitting eld of
(x2 2)(x2 5)(x2 7) over Q. By the Fundamental Theorem of Galois Theory K Gal(E/Q)
describes a bijective correspon
MATH 431
Written Homework 11 Solution
Radford
04/21/2009
1. Page 540, number 2: (30 points) 2 (10), 3 (10), 3 (10) respectively.
2. Page 540, number 4: (30 points) Write u = (a1 , . . . , an ) and v = (b1 , . . . , bn ). Then
d(u, v ) = |cfw_i | 1 i n, ai
MATH 431
Written Homework 9 Solution
Radford
04/09/2009
1. Page 415, number 36: (30 points) H is a normal subgroup of a nite group G
and |H | = p for some positive prime p and 0. We may assume that > 0. By
Sylows Second Theorem H K for some Sylow p-subgro