Math 323: Homework 6
Problem 9.1 Problem 4. Prove that the ideals (x) and (x, y) are prime ideals
in Q[x, y] but only the latter ideal is a maximal ideal.
Solution: We rst consider the map
: Q[x, y] Q[y]
dened by (f (x, y) = f (0, y) for every polynomial
Math 323: Homework 4
1. Find the necessary and sucient condition on the integers D1 and D2 for the
elds Q( D1 ) and Q( D2 ) to be isomorphic.
Solution. The answer is: D1 and D2 dier by a square, i.e., there exists c Q
such that D1 = c2 D2 .
and so any ele
Math 323: Homework 5
Section 8.1 Problem 10. Prove that the quotient ring Z[i]/I is nite for every
ideal I.
Solution: As the hint suggests, note that every ideal is principal, since it is a
Euclidean ring, so I = () for some Z[i]. Then, since we know that
1
Pells equation and continued fractions
This is a series of (optional) problems about Pells equation:
x2 Dy 2 = 1
(1)
that leads to understanding the set of its solutions, and therefore, of the units
in the ring Z[ D]. It does not use any sophisticated t
Math 323: Solutions to homework 9
Problem 1. (a) Let R be an integral domain, and let M be an R-module. Prove
that if Tor(M ) = cfw_0, then M cannot be free on any set of generators.
Note that as proved in the last homework, if R has zero divisors, then a
Math 323: Homework 7
Section 13.1 Problem 1. Show that p(x) = x3 + 9x + 6 is irreducible in Q[x].
Let be a root of p(x). Find the inverse of 1 + in Q().
Solution: Applying Eisensteins criterion for prime p = 3, we see that x3 +9x+6
is irreducible in Z[x],
Math 323: Solutions to Homework 8
1. Let F be a eld, and V1 , V2 be nite-dimensional F -vector spaces. Prove that
the F [x]-module (V1 , T1 ) is isomorphic to the F [x]-module (V2 , T2 ) if and only if
dim V1 = dim V2 and there exists a basis cfw_1 , . .
Math 323. Midterm Exam. February 28, 2013. Time: 75 minutes.
(1) [2 4 = 8]. Are the following statements true or false? (Please give the
answer with brief justication/counterexample in each case; you do not
need to include complete details, just sketch th
Math 323. Midterm Exam. February 28, 2013. Time: 75 minutes.
(1) [2 4 = 8]. Are the following statements true or false? (Please give the
answer with brief justication/counterexample in each case; you do not
need to include complete details, just sketch th
Problem Set 10 (and last) all these problems are for extra credit; you
do not have to turn it in. Due Tuesday April 8; no extensions: solutions
will be posted right away.
1. Section 10.3, Problem 17.
Solution. (We actually did this in class.)
Consider the
List of topics for the Midterm
(1) The denitions of a ring, commutative ring, ring with identity, homomorphism and isomorphism of rings.
(2) Key examples: the quaternions, quadratic integer rings, polynomial rings,
rings of functions on a set, matrix ring
Some more (optional) problems for Math 323.
This is a compilation of both basic exercises and harder problems that are useful/interesting, but were not included in the homework; many of these are from
Dummit and Foote. Solve as many as you like. Please do
REMARKS ABOUT EUCLIDEAN DOMAINS
KEITH CONRAD
1. Introduction
The following denition of a Euclidean (not Euclidian!) domain is very common in
textbooks. We write N for cfw_0, 1, 2, . . . .
Denition 1.1. An integral domain R is called Euclidean if there is
Math 323. Practice problems on modules.
(1) Let R be an integral domain, and let F be a free module over R of nite rank, and let R2 be the free module of rank 2 over R. Prove that
HomR (F, R2 ) F F .
Solution. This problem is basically the same as the ear
Math 323: Extra problems of Number-theoretic avour
1. Find all integer solutions to y 2 + 1 = x3 with x, y = 0 (Hint: use the ring of
Gaussian integers Z[i]).
2. Now we can revisit Pells equation 2 2y 2 = 1.
x
(a) Show that there is no unit in Z[ 2] R suc
Math 323. Midterm Exam. February 27, 2014. Time: 75 minutes.
(1) [5] Find the centre of the ring of real quaternions H.
Answer: R = cfw_a + 0i + 0j + 0k | a R. It is clear that all real
elements commute with everything. To prove that no other elements are
Math 323: Homework 2 solutions
Section 7.2 Problem 3. Dene the set R[x] of formal power series in the
indeterminate x with coecients from R to be all formal innite sums
n=0
an xn = a0 + a1 x + a2 x2 + a3 x3 +
Dene addition and multiplication of power ser
THE UNIVERSITY OF BRITISH COLUMBIA
SESSIONAL EXAMINATIONS APRIL 2013
MATHEMATICS 323
Time: 2 hours 30 minutes
Instructions: You can use the statements we proved in class, or the theorems
proved in the textbook, without proof (except the question 4e); but
List of topics for the Math 323 nal exam, April 12 2014.
The exam will be approximately 2/3 on rings, with emphasis on irreducibility
of polynomials and construction of eld extensions (the things that happened after
the midterm) and 1/3 on modules. It cov
THE UNIVERSITY OF BRITISH COLUMBIA
SESSIONAL EXAMINATIONS APRIL 2013
MATHEMATICS 323
Time: 2 hours 30 minutes
Instructions: You can use the statements we proved in class, or the theorems
proved in the textbook, without proof (except the question 4e); but
Math 323: Homework 3
Section 7.3 Problem 29. Let R be a commutative ring. Recall that an
element x R is nilpotent if xn = 0 for some n Z+ . Prove that the set of
nilpotent elements form an ideal - called the nilradical of R and denoted by N(R).
Solution:
Problem Set 1. Due Thursday January 16.
1. Prove, without using the Fundamental Theorem of Arithmetic, that an integer
p is prime if and only if it has the following property: for any two integers a, b, if
p|ab then p|a or p|b.
Solution. The denition of a
Math 323. Practice problems on modules.
(1) Let R be an integral domain, and let F be a free module over R of nite rank, and let R2 be the free module of rank 2 over R. Prove that
HomR (F, R2 ) F F .
(2) Let R be an integral domain, and let M be a torsion