WORKSHEET # 8
IRREDUCIBLE POLYNOMIALS
We recall several dierent ways we have to prove that a given polynomial is irreducible. As always, k is a eld.
Theorem 0.1 (Gauss Lemma). Suppose that f Z[x] is monic of degree > 0. Then f is irreducible in Z[x] if
an
QUIZ #1 MATH 435
FEBRUARY 6TH, 2012
1. Suppose that G is a cyclic group of order 10 with generator a G. Write down all of the
elements of G AND identify the order of each element of G.
Solution: The elements of G are cfw_e, a1 , a2 , a3 , a4 , a5 , a6 , a
HOMEWORK #9 MATH 435
SOLUTIONS
Chapter 4, Section 6: #2 Prove that f (x) = x3 + 3x + 2 is irreducible in Q[x].
Solution: By Gauss Lemma, it is sucient to show that this is irreducible in Z[x]. Now, note
that f (x) is irreducible if and only if f (x + 1) i
HOMEWORK #8 MATH 435
SOLUTIONS
Chapter 4, Section 5: #1 If F is a eld, show that the only invertible elements in F [x] are the
non-zero elements of F .
Solution: Certainly the elements of F are invertible. Conversely, suppose that g F [x] is
invertible bu
HOMEWORK #7 MATH 435
SOLUTIONS
Chapter 4, Section 3: #1 If R is a commutative ring and a R, let L(a) = cfw_x R | xa = 0.
Prove that L(a) is an ideal of R.
Solution: Indeed, we need to show that L(a) is a subgroup under addition, and closed under
multiplic
HOMEWORK #6 MATH 435
DUE MONDAY MARCH 19TH
Chapter 4, Section 1: #35 For R as in Example 10, show that S = cfw_f R | f is dierentiable on (0, 1)
is a subring of R which is not an integral domain.
Solution: First we need to prove that S is a subring. It is
SOME SOLUTIONS TO HOMEWORK #5
MATH 435 SPRING 2012
Certainly there are many correct ways to do each problem.
#2 on page 87. Let G be the group of all real-valued functions on the unit interval [0, 1],
where we dene, for f, g G, addition by (f + g )(x) = f
SOME SOLUTIONS TO HOMEWORK #3
MATH 435 SPRING 2012
Certainly there are many correct ways to do each problem.
#21 on page 75. Let S be any set having more than two elements. If s S , we dene
Hs = cfw_f A(S )|f (s) = s.
. Prove that Hs cannot be a normal su
SOME SOLUTIONS TO HOMEWORK #2
MATH 435 SPRING 2012
Certainly there are many correct ways to do each problem.
#4 on page 54. Verify that Z (G), the center of G, is a subgroup of G.
Proof. First note that ea = a = ae for all a G which proves that e Z (G). N
SOME SOLUTIONS TO HOMEWORK #1
MATH 435 SPRING 2012
Certainly there are many correct ways to do each problem.
#2 from page 50. Suppose G is a nite set with a associative binary operation satisfying the
rule ab = ac implies that b = c and also that ba = ca
QUIZ #2 MATH 435
FEBRUARY 15TH, 2012
1. Consider the group U (9) (the group of positive integers less than and relatively prime to 9 under
multiplication mod 9) and the cyclic subgroup H = 8 .
(a) Explain why H is a normal subgroup, write down the element
QUIZ #3 MATH 435
MARCH 12TH, 2012
1. Consider the following elements of S4 :
= (123)
= (12)(34)
Compute 1 and also compute 1 (by compute, we mean disjoint cycle form, or as a cycle).
What do you notice about the shape of the outputs? (2 points)
Solution
WORKSHEET # 8
IRREDUCIBLE POLYNOMIALS
We recall several dierent ways we have to prove that a given polynomial is irreducible. As always, k is a eld.
Theorem 0.1 (Gauss Lemma). Suppose that f Z[x] is monic of degree > 0. Then f is irreducible in Z[x] if
an
WORKSHEET # 7
QUATERNIONS
The set of quaternions are the set of all formal R-linear combinations of symbols i, j, k
for a, b, c, d R.
a + bi + cj + dk
We give them the following addition rule:
(a + bi + cj + dk ) + (a + b i + c j + d k ) = (a + a ) + (b +
WORKSHEET # 4 SOLUTIONS
MATH 435 SPRING 2011
We rst recall some facts and denitions about cosets. For the following facts, G is a group and H
is a subgroup.
(i) For all g G, there exists a coset aH of H such that g aH . (One may take a = g ).
(ii) Cosets
WORKSHEET # 4
MATH 435 SPRING 2012
We rst recall some facts and denitions about cosets. For the following facts, G is a group and H
is a subgroup.
(i) For all g G, there exists a (left) coset aH of H such that g H . (One may take a = g ).
(ii) Cosets are
WORKSHEET # 3 (RSA CRYPTOGRAPHY)
MATH 435 SPRING 2011
Consider the group U (n), the set of integers between 1 and n 1 relatively prime to n, under
multiplication mod n.
1. Suppose that p and q are distinct primes. What is the order of U (pq ), |U (pq )|?
WORKSHEET # 3 (RSA CRYPTOGRAPHY)
MATH 435 SPRING 2011
Consider the group U (n), the set of integers between 1 and n 1 relatively prime to n, under
multiplication mod n.
1. Suppose that p and q are distinct primes. What is the order of U (pq ), |U (pq )|?
WORKSHEET #2
The following 8 transformations make up the group D4.
1
WORKSHEET #2
2
You may assume that D4 is indeed a group (for now.).
1. Prove that D4 is not Abelian.
Solution: f 1 f 2 = do f2 rst then f1 = r90 but f 2 f 1 = do f1 rst then f2 = r270.
3
WORKSHEET #1 (IDENTIFYING GROUPS) MATH 435
Below are sets with potential binary operations (ie, a way to combine elements). Determine if each set is
(or is not) a group and prove your answer. If it is a group, is it Abelian?
1. For a xed integer n, the nu
QUIZ #4 MATH 435
MARCH 23RD, 2012
1. Suppose that R is a ring.
(a) Prove that a(b) = (ab) for all a, b R. (1 point)
(b) Suppose R is a commutative ring for which ab = ac implies b = c whenever a = 0. Prove
that R is an integral domain. (1 point)
Solution:
GROUPS ACTING ON A SET
MATH 435 SPRING 2012
NOTES FROM FEBRUARY 27TH, 2012
1. Left group actions
Denition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a
rule for combining elements g G and elements x S , denoted by g.x
MATH 435, FINAL EXAM
Your Name
You have 2 hours to do this exam.
No calculators!
No notes!
For proofs/justications, please use complete sentences and make sure to explain any steps which
are questionable.
All rings will be assumed to be commutative, asso
HOMEWORK #9 MATH 435
SOLUTIONS
Chapter 4, Section 6: #2 Prove that f (x) = x3 + 3x + 2 is irreducible in Q[x].
Solution: By Gauss Lemma, it is sucient to show that this is irreducible in Z[x]. Now, note
that f (x) is irreducible if and only if f (x + 1) i
HOMEWORK #8 MATH 435
SOLUTIONS
Chapter 4, Section 5: #1 If F is a eld, show that the only invertible elements in F [x] are the
non-zero elements of F .
Solution: Certainly the elements of F are invertible. Conversely, suppose that g F [x] is
invertible bu
HOMEWORK #7 MATH 435
SOLUTIONS
Chapter 4, Section 3: #1 If R is a commutative ring and a R, let L(a) = cfw_x R | xa = 0.
Prove that L(a) is an ideal of R.
Solution: Indeed, we need to show that L(a) is a subgroup under addition, and closed under
multiplic
HOMEWORK #6 MATH 435
DUE MONDAY MARCH 19TH
Chapter 4, Section 1: #35 For R as in Example 10, show that S = cfw_f R | f is dierentiable on (0, 1)
is a subring of R which is not an integral domain.
Solution: First we need to prove that S is a subring. It is
SOME SOLUTIONS TO HOMEWORK #5
MATH 435 SPRING 2012
Certainly there are many correct ways to do each problem.
#2 on page 87. Let G be the group of all real-valued functions on the unit interval [0, 1],
where we dene, for f, g G, addition by (f + g )(x) = f
SOME SOLUTIONS TO HOMEWORK #3
MATH 435 SPRING 2012
Certainly there are many correct ways to do each problem.
#21 on page 75. Let S be any set having more than two elements. If s S , we dene
Hs = cfw_f A(S )|f (s) = s.
. Prove that Hs cannot be a normal su
SOME SOLUTIONS TO HOMEWORK #3
MATH 435 SPRING 2012
Certainly there are many correct ways to do each problem.
#28 on page 65. If G is a cyclic group of order n, show that there are (n) generators for
G. Give their form explicitly.
Proof. Suppose that G = a