Pre-class worksheet due Wednesday, November 16 (Solutions).
1. Consider the elements a = 8 + 18i and b = 2 + 3i in the ring Z[i] of Gaussian integers. Recall that Z[i] is a Euclidean ring with the norm N (x + iy) = x2 + y 2 where x, y Z. Note: There was a
Pre-class worksheet due Monday, November 14, 2011 (Solutions).
1. For each of the following statements indicate if it is true or false:
(1) The ring C is a Euclidean domain.
(2) The ring Z is a Euclidean domain.
(3) The ring Z[x] is a Euclidean domain.
(4
Pre-class worksheet due Wednesday, November 9, 2011 (Solutions).
1. For each of the following statements indicate if it is true or false:
(1) The ideal 5Z Z is a maximal ideal in Z.
(2) There does not exist a eld F such that F has four distinct ideals.
(3
Pre-class worksheet due Monday, November 7 (Solutions).
1. Consider the ring Z and the ideals I = 6Z, J = 4Z in Z.
Taking for granted that every ideal in Z has the form nZ for some n Z,
represent in this form each of the following ideals:
I + J, IJ, I 2 ,
Pre-class worksheet due Friday, November 4 (Solutions).
1. Consider the function : R[x] C given by the formula (f ) = f (i) for every polynomial f (x) R[x]. More specifically, for every f (x) = a0 + a1 x + + an xn R[x] we have (f ) := a0 + a1 i + a2 i2 +
Pre-class worksheet due Wednesday, November 2. (Solutions)
1. For each of the following statements indicate if it is true or false.
(1) If R, S are rings with 1 and : R S is a ring homomorphism then
(1) = 1.
(2) If R, S are rings with 1 and : R S is a rin
Math 417 Exam 2 (Solutions) Prof. I.Kapovich November 11, 2011 Problem 1.[20 points] For each of the following statements indicate whether it is true or false. You DO NOT need to provide explanations for your answers in this problem. (1) If G is a group a
Math 417 Section C1 Exam 1 (Solutions) Prof. I. Kapovich September 30, 2011 Problem 1.[20 points] For each of the following statements indicate whether it is true or false. You DO NOT need to provide explanations for your answers in this problem. (1) The
Pre-class worksheet due Monday, December 5, 2011 (Solutions).
1. For each of the following statements indicate if it is true or false:
(1) The ring Q[x]/(x4 + 6x3 3x2 + 9x + 3) is a eld.
(2) The ring R[x]/(x2 + 2) is a eld.
(3) The ring C[x]/(x2 + 2) is a
Pre-class worksheet due Friday, December 2, 2011 (Solutions).
1. For each of the following statements indicate if it is true or false:
(1) The polynomial x4 + 6x3 3x2 + 9x + 3 is irreducible in Z[x].
(2) The polynomial x4 + 6x3 3x2 + 9x + 3 is reducible i
Pre-class worksheet due Wednesday, Aug 31 (solutions).
PRINT YOUR NAME:
1. Consider the set (R, +) as a group with respect to addition.
Compute the 3-d power of 5 R with respect to addition.
What is the order of the element 5 in (R, +)?
Answer:
The 3-d po
Pre-class worksheet due Monday, Aug 29 (Solutions).
1. Consider the set R with the operation x
(a) Compute (3 2) 5 and 3 (2 5).
y := xy + 1, where x, y R.
Solution. We have:
(3
2)
5 = (6 + 1)
5=7
5 = 36
(2
11 = 34.
and
3
5) = 3
(b) Is the operation on R a
Pre-class worksheet due Friday, Aug 26 (solutions).
1. Compute the following elements in Z/6Z (that is, represent them in
the form a, where 0 a 5):
(a)
(3 5) + 13 = 15 + 13 = 28 = 4.
(b)
7100 = (7)100 = 1
100
= 1.
2. Does the equation 3x = 1 have a soluti
Pre-class worksheet due Wednesday, September 21 (Solutions).
1. Let G be a group such that Z (G) = G. Does this imply that G is
abelian?
Suppose now that G is an abelian group. Does this imply that Z (G) = G?
Answer:
If Z (G) = G then, yes, the group G is
Pre-class worksheet due Wednesday, September 28 (Solutions).
1. For each of the following statements indicate whether it is true or false:
(1) For every group G, every subgroup H G and every g G we have
gH = Hg .
(2) If H G and g1 , g2 G then either g1 H
Pre-class worksheet due Monday, September 26 (Solutions).
1. For each of the following statements indicate whether it is true or false:
(1) For every group G we have G = G .
(2) Every abelian group can be generated by a subset consisting of a
single eleme
Pre-class worksheet due Friday, September 23 (Solutions).
1. List all cyclic subgroups of the group S3 . How many distinct cyclic
subgroups does S3 have?
Solution. The cyclic subgroups of S3 are:
1 = cfw_1
(1 2) = cfw_1, (1 2)
(1 3) = cfw_1, (1 3)
(2 3) =
Pre-class worksheet due Wednesday, September 14 (Solutions)
1. Consider the map
exp : R (0, ),
exp(x) = ex .
Is it true that the map exp from the group R with respect to addition to
the group (0, ) with respect to multiplication is an isomorphism?
Answer: