Mathematics 111
Spring 2009
Homework 5
1. Let R be a PID and M a nitely generated R-module. Show that M is projective if
and only if it is free.
2. Let M be a submodule of Zn having group index p in M , i.e., [Zn : M ] = p, where p is
a prime. Show that M
Dartmouth College
Mathematics 81/111 Homework 2
In the rst couple of exercises, we explore the notions of prime and maximal ideals in
non-commutative rings. We start with two basic denitions. Here A is a general ring which
does not necessarily have an ide
Dartmouth College
Mathematics 81/111 Homework 2
In the rst couple of exercises, we explore the notions of prime and maximal ideals in
non-commutative rings. We start with two basic denitions. Here A is a general ring which
does not necessarily have an ide
Dartmouth College
Mathematics 81/111 Homework 1
Some basic denitions concerning algebraic sets
Let k be a eld, and let k [x1 , . . . , xn ] the polynomial ring in n variables with coecients
in k . For f k [x1 , . . . , xn ] and P = (a1 , . . . , an ) k n
Dartmouth College
Mathematics 81/111 Homework 3
1. A rst approximation to a theorem of Bezout.
(a) Let A be a UFD with eld of fractions F . Let f, g A[x]. Show that f, g are
relatively prime in A[x] if and only if f, g are relatively prime in F [x] and (t
Dartmouth College
Mathematics 81/111 Homework 3
1. A rst approximation to a theorem of Bezout.
(a) Let A be a UFD with eld of fractions F . Let f, g A[x]. Show that f, g are
relatively prime in A[x] if and only if f, g are relatively prime in F [x] and (t
Math 81/111 Midterm Exam
6 February 2014
Your name (please print):
Please attach this cover sheet to your solutions to the exam.
Instructions: Your solutions to this exam are due in class on Friday, 14 February 2014
at 10:00 am. You may feel free to use s
Dartmouth College
Mathematics 81/111 Homework 5
1. Let L/K be an extension of elds. We say that K is algebraically closed in L if the only
elements of L which are algebraic over K are the elements of K . Let x be transcendental
over K .
(a) Show that K is
Dartmouth College
Mathematics 81/111 Homework 4
1. F be a eld of characteristic 0, and let m and n be distinct integers with
Let
n F , and mn F .
/
/
(a) Show that [F ( m, n) : F ] = 4.
m F,
/
(b) Show by example
(with m/n )2 ) that the above statement c
Dartmouth College
Mathematics 81/111 Homework 4
/
1. F be a eld of characteristic 0, and let m and n be distinct integers with m F ,
Let
n F , and mn F .
/
/
(a) Show that [F ( m, n) : F ] = 4.
Solution: Since m F and x2 m F [x] has m as roots, we
/
have
Mathematics 111
Spring 2009
Homework 1
1. For a positive integer m, let Z/mZ denote the usual ring of integers modulo m. We
wish to consider the existence of ring homomorphisms : Z/mZ Z/nZ and their
properties. Note that this will also inform us of proper
Mathematics 111
Spring 2009
Homework 2
1. Let R be a ring with identity. Show that the sequence of left R-modules
0
/L
/M
/N
is exact if and only if for all left R-modules D, the sequence
0
/ HomR (D, L)
/ HomR (D, M )
/ HomR (D, N )
is exact.
Hint: We ha
Mathematics 111
Spring 2009
Homework 3
1. (Commutative diagrams gone mad) Given a ring R with identity and R-modules
A, B, M , consider the following diagram with R-linear maps f, g :
B
g
/M
f
A
A pullback for this diagram (also called a ber product of f
Mathematics 111
Spring 2009
Homework 4
1. (A theorem not proven in class). Let R be a ring with identity, M a right R-module
and N a free left R-module with basis cfw_ei iI . Show that every element in M R N
can be written uniquely as a nite sum iI mi ei
Dartmouth College
Mathematics 81/111 Homework 1
Some basic denitions concerning algebraic sets
Let k be a eld, and let k [x1 , . . . , xn ] the polynomial ring in n variables with coecients
in k . For f k [x1 , . . . , xn ] and P = (a1 , . . . , an ) k n