Solutions to the selected problems (Homework 57)
Linear Algebra
Fall 2010
t + x y + iz
.
y iz t x
Page 86, 3) Let T be the given function, so T (x, y, z, t) =
Then
T (c(x, y, z, w) + (x , y , z , w ) = T (cx + x , cy + y , cz + z , cw + w )
=
ct + t + cx
Math 5032, Algebra II
Problem Set 2
Due: February 23 in class
1. Let F be a nite eld, and let E be a nite extension of F . Show that E/F is
Galois and the Galois group is cyclic.
2. Let p be a prime number and let E = Fp (x, y) and F = Fp (xp , y p ). Sho
Math 5032, Algebra II
Problem Set 3
Due: March 16 in class
1. Let A B be integral domains, and let C be the integral closure of A in B.
Let f (x) and g(x) be monic polynomials in B[x] such that f g C[x]. Show that
f, g C[x] (Consider a eld containing B in
Math 5032, Algebra II
Problem Set 7
Due: May 4
In Problems 1-4, K is a number eld with ring of integers OK .
1. Find the ring of integers of Q( d) where d is a square free integer and d
3 (mod 4) .
2. Suppose that A B is a ring extension. Let M = 0 be a
Math 5032, Algebra II
Problem Set 5
Due: April 13 in class
All the rings are commutative.
1. (a) Let M be an A-module. Show that M = 0 if and only if Mm = 0 for every
maximal ideal m of A.
(b) Show that if I J A are two ideals such that for each maximal i
Solutions to the selected problems (Homework 12)
Linear Algebra
Fall 2010
Page 289, 6) (a) Let = (3, 4). By Theorem 4 of page 284, E() which is
the same as the best approximation to in W , is given by E() = (|) .
|2
Since | = 5, if = (x1 , x2 ), we have
E
Solutions to the selected problems (Homework 10)
Linear Algebra
Fall 2010
Page 198, 7) B = cfw_1, x, x2 , . . . , xn is a basis for V . The matrix of D with
respect to B is lower triangular, and all the diagonal entries are zero, so
the characteristic po
Solutions to the selected problems (Homework 11)
Linear Algebra
Fall 2010
noindent Page 275, 2). If (|) and < , > are two inner product,
then the sum and the dierence of them is linear with respect to the rst
component, and
(|) < | >= (|) < | > = (|) < |
Solutions to the selected problems (Homework 8,9)
Linear Algebra
Fall 2010
Page 163, 8) Since TB (I ) = IB BI = 0, the nullspace of TB is not the
zero space, therefore TB is singular and not invertible, so det TB = 0.
Page 190, 8) To show that I BA is inv