Graduate Algebra
Homework 8
Fall 2014
Due 2014-11-12 at the beginning of class
Throughout this problem set R is a commutative ring. Recall that p R is a prime ideal if p = R and
R/p is an integral domain, and m R is a maximal ideal if m = R and R/m is a e
Graduate Algebra
Homework 3 Solutions
Fall 2014
Due 2014-09-17 at the beginning of class
1. (a) Show that Aut(Q) Q .
=
(b) Show that Aut(R)
R . [Hint: Take a suitable Q-vector space projection from R to Q.]
(c) (Extra credit) Find all groups G such that A
Graduate Algebra
Homework 4
Fall 2014
Due 2014-09-24 at the beginning of class
1. Recall the quaternion group from homework 2.
(a) Show that Q has the following presentation: Q i, j|i2 = j 2 = (ij)2 .
=
(b) Deduce that | Aut(Q)| = 24. [Hint: Make a list o
Graduate Algebra
Homework 5
Fall 2014
Due 2014-10-01 at the beginning of class
1. Let n 5.
(a) Show that the only proper normal subgroup of Sn is An .
(b) Let H be a proper subgroup of Sn . Show that either H = An or [Sn : H] n. [Hint: Consider
the action
Graduate Algebra
Homework 6
Fall 2014
Due 2014-10-08 at the beginning of class
ab
1. Show that Sn Z/2Z by showing that [Sn , Sn ] = An .
=
Proof. Know that [G, G] G and so [Sn , Sn ] is one of 1, An or Sn by the last homework. Its not 1
since Sn is not ab
Graduate Algebra
Homework 7
Fall 2014
Due 2014-10-29 at the beginning of class
1. Let N be a normal subgroup of G. If N and G/N are solvable, show that G is solvable.
Proof. The solvability condition implies that N = N0 N1 . . . Nk = 1 with Ni /Ni+1 abeli
Exam 3D solutions
Multiple choice.
(1) Separating variables in
dy
dx
=
1x2
y
gives y dy = (1 x2 ) dx so
y dy = (1 x2 ) dx, 1 y 2 = x 1 x3 + C and y = 2x 2 x3 + 2C.
2
3
3
When x = 0, y(0) = 4 = 2C so C = 8 and the sign is +. Thus,
(x) = 2x 2 x3 + 16. So (3
2
Initials:
1.(6pts) Let A be an n n matrix satisfying AT A = I. Let u, v be vectors in Rn such that
u v = 4. Find (Au) (Av).
(a) 1/4
(b) 1/4
(c) 0
(d) 4
() 4
Solution: T
A A = I means A is unitary so (Au) (Av) = u v = 4.
1
1
1
1 1 1
,
,
. Compute
w.
I). W ¢ 6W).
, - 1 - 0,40,
e/ZK M 2 0k}ka J> m, aw/gm n
00/; ,L/Z)/$(@) M
73 m e 5 ~
H +F MM?
7ww£wkt WM ' f m {M
67w
.40 m 'wbl # 0
LA K
L/K 'Tm/K )NLK (9
(a) (6(%&r;4+ LVTA +°1 m) / 201?be
TrK/@(rm) 1
.; Zap/«dd:
/@(\/Km7 WWW)
I ZERO A #g
6 (96mm
Graduate Algebra
Homework 9
Fall 2014
Due 2014-11-19 at the beginning of class
1. Let R be a PID. Throughout this exercise, R represents a prime element, P (X) R[X] is an
irreducible polynomial and Q(X) R[X] is a polynomial whose image in R/()[X] is irred
Graduate Algebra
Homework 2
Fall 2014
Due 2014-09-10 at the beginning of class
1. Show that the dihedral group D8 with 8 elements is isomorphic to the subgroup of GL(2, R) generated
0 1
0 1
by the matrices
and
.
1 0
1 0
Proof. First, remark that if G is a