MATH 251: ABSTRACT ALGEBRA I
EXAM #3
Problem 1. Let F be a eld. For a polynomial f (x) =
n
i=0
ai xi F [x], we denote by
n
iai xi1 = a1 + 2a2 x + + nan xn1
f (x) =
i=1
the formal derivative of f .
(a) Let
I = cfw_f (x) F [x] : f (1) = f (1) = 0.
Show that
MATH 321 Final Examination Answer Key
(1) Give an example of a fact that you have known long before taking MATH 321
but which you recognized in MATH 321 to be a piece of information about groups.
There are lots of possible answers here. For example, you m
Solutions to Problem Session Questions
Problem: (3.3) Let f : A B be a function. Prove that f is one-to-one
if and only if f (A1 ) f (A2 ) = f (A1 A2 ) for any subsets A1 , A2 of A.
We know from Ex. 2 that f (A1 A2 ) f (A1 ) f (A2 ). The claim amounts
to
HOMEWORK 1 SOLUTIONS
MATH 121
Problem (10.1.2). Prove that
R
and
M
group action of the multiplicative group
satisfy the two axioms in Section 1.7 for a
R
on the set
M.
Solution. For the rst axiom, we have to check that for
r1 , r2 R
and
m M,
we
have
For t
Math 322 Final Examination
Name :
Signature :
May 31, 1999
9:0012:00
(1) Show that a polynomial f (x) Z[x] is irreducible in Z[x] if and only if f (x + 1)
is irreducible in Z[x]. Using this, prove that for any odd prime number p, the pth
cyclotomic polyno
Math 322 First Midterm Examination
Name :
April 8, 1999
Signature :
9:0010:30
(1) Is 3x6 4x + 6 irreducible in Z[x] ?
(10 points)
(2) Does there exist a polynomial f in Z[x] such that f (0) = 1, f (1) = 2, f (2) = 3,
f (3) = 2 ?
(20 points)
Name :
Surname