The University of British Columbia
Practice Midterm Examination - February, 2016
MATH 323
Closed book examination
Time: 75 minutes
Last Name
First
Signature
Student Number
Special Instructions:
No memory aids are allowed. No calculators may be used. Show all your work; little or no
credit will be given for a numerical answer without the correct accompanying work. If you
need more space than the space provided, use the back of the previous page. Where boxes
are provided for answers, put your final answers in them.
Rules governing examinations
• Each candidate must be prepared to produce, upon request, a
UBCcard for identification.
• Candidates are not permitted to ask questions of the invigilators,
except in cases of supposed errors or ambiguities in examination
questions.
• No candidate shall be permitted to enter the examination room
after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination.
• Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination
and shall be liable to disciplinary action.
(a) Having at the place of writing any books, papers
or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness
shall not be received.
• Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any
examination material from the examination room without permission of the invigilator.
• Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.
Page 1 of 6 pages
1
10
2
10
3
10
4
10
5
10
Total
50
February, 2016
Math 323
Name:
Page 2 of 6 pages
1. Let R be a commutative ring with 1. Prove that if there exists a prime ideal P of R that
contains no zero divisors, then R is an integral domain.
February, 2016
Math 323
Name:
Page 3 of 6 pages
2. Let R be a commutative ring with 1, and I, J – ideals in R.
(a) Prove that for any two ideals I, J in a commutative ring R, IJ ⊆ I ∩ J.
(b) Give a sufficient condition for the equality IJ = I ∩ J to hold (just a statement, no
proof required).
(c) Give an example of two ideals I and J in a commutative ring R, such that IJ 6= I ∩ J.
February, 2016
Math 323
Name:
Page 4 of 6 pages
3.
√
(a) Is 7 prime in Z[ 1+ 2 −3 ]? If not, factor it as a product of primes, with proof that the
factors are prime.
√
(b) Find an example of an element of Z[ −3] that is irreducible but not prime (and give
a complete proof that it has this property).
February, 2016
Math 323
Name:
Page 5 of 6 pages
4. Describe the quotient ring (i.e. find a simpler-looking ring isomorphic to it). Is the ideal
(x2 + 1) maximal in either of these rings?
(a) (Z/5Z)[x]/(x2 + 1)
(b) (Z/7Z)[x]/(x2 + 1).
February, 2016
Math 323
Name:
Page 6 of 6 pages
5. Let F be a field that has infinite cardinality. Let n be an arbitrary integer. Prove that
for any collection of elements a1 , . . . an ∈ F , and any collection of values c1 , . . . , cn ∈ F there
exists unique polynomial f ∈ F [x] of degree at most n − 1 such that f (ai ) = ci for 1 ≤ i ≤ n.