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 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 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
The Forty-Niners
The California Gold
Rush
The World
Rushes In
2/3 of 49ers were
Americans, rest
from Mexico,
South America,
Europe, Australia,
and China
Most were young
and male
Life in the Mining Camps
No police, camps rough
Miners took it on themsel