PMath 334 Assignment 1 Solutions
1. Are the following subsets of C rings with the usual operations? If they are rings, are they
elds? Since they are subsets of C and we care about the usual ring operations, you may use the
subring test.
(a) R = cfw_ a Q :
PMath 334 Assignment 6 Solutions - Due in class Wednesday, June 27, 2012. Hand in solutions
to the following problems. Be sure to justify your answers and neatly present your work.
1. Let R be a ring and let a, b R. Prove that (a)(b) = (ab).
Solution: An
PMath 334 Assignment 5 Solutions.
1. Find the (monic) gcd of f (x) and g (x) in k [x] and write it as uf + vg . Is f a unit of k [x]/(g )?
If so, what is its multiplicative inverse?
(a) f = x2 5, g = x4 + 1, k = Q
Solution: Long division yields that g = (
PMath 334 Assignment 4 - Solutions
Neatly present your work and justify your answers for the following problems.
1. Let R be a ring and let b R. We say that b is idempotent if b2 = b. For example, 0 and 1
are idempotents. Find a non-trivial (meaning not 0
PMath 334 Assignment 2 - Due in class Wednesday, May 23, 2012.
Hand in solutions to the following problems - be sure to neatly present your work and to justify
your answers.
1. The ring Z4 does not contain a square root of 2, since 02 = 0, 11 = 1, 22 = 0,
PMath 334 Assignment 3 Solutions - Due in class Wednesday, May 30, 2012.
Neatly present your work and justify your answers for the following problems.
1. Find all of the ideals of Z12 . For each ideal F which isnt all of Z12 , nd an integer N Z so
that Z1
PMath 334 Assignment 7 Solutions. Hand in solutions to the following problems. Be sure to
justify your answers and neatly present your work.
1. Let = 2+ 2i C. Find the minimal polynomial of over the following elds k . Be sure
2
to explain why your polynom
PMath 334 Assignment 8 Solutions - Due in class Wednesday, July 18, 2012. Hand in solutions
to the following problems. Be sure to justify your answers and neatly present your work.
1. Let R be an integral domain and let a, b R. Prove that (a) = (b) if and
PMath 334 Quiz 3 Solutions - Friday, May 25, 2012
Write your name and ID number somewhere at the top of the page. Justify all of your answers. Each of the three questions are of equal value.
1. The third condition that a subset I of R is required to satis
PMath 334 Quiz 4 - Friday, June 1, 2012
Write your name and ID number somewhere at the top of the page. Justify all of your answers. Each of the two questions are of equal value.
1. Prove that R[x]/(x2 + 9) C. I suggest that you write down a homomorphism
PMath 334 Quiz 2 - Friday, May 18, 2012
Write your name and ID number somewhere at the top of the page. Each of the three questions
are of equal value.
1. Let C satisfy 2 = + 3. Consider the function : Z[t] Q[ ] (here Z[t] is the
ring of polynomials in t
PMath 334 Quiz 1 Solutions
Write your name and ID number somewhere at the top of the page.
1. True or False? Just say your answer, no need to justify anything!
(a) [6] is a unit of Z15 . Answer: False
Solution 1: [6] is a zero-divisor of Z15 since [6][5]
PMath 334 Assignment 9 - Not due at all, but good practice. These are not neccesarily in order
of increasing diculty.
1. (In honour of the last quiz question which was poorly stated, but not poorly stated if you
know this result). Let k be a subeld of C.
PMath 334 Midterm Test Solutions - Monday, June 11, 2012. By a ring I mean a commuting
ring where 0 = 1. All homomorphisms send 1 to 1.
1. Yes or no? Just write down your answer in your booklet!
[2](a) Is Q[x] a eld?
Solution: No - any degree 1 or higher