Dr. Martin Olbermann
Due: Wednesday, Mar. 12, in class
Math 114
Spring 2008
Homework 7
Proofs and explanations should always be written using complete English sentences. You should always explain and
Partial Solutions to Homework 6
1. Let K be a field of characteristic p > 0.
a) The Frobenius map is the map p : K K mapping a 7 ap . Show that it is an injective
homomorphism.
b) Prove that if K is a
Solutions to Homework 12
1. Let L : K be a normal and separable field extension of degree p. In this exercise we shall
see from a different viewpoint (namely the one of linear algebra) without the use
Solutions to Homework 4
1. Stewart, exercise 3.9.
Solution:
Q()
= Q[t]/ht2 2i and Q()
= Q[t]/ht2 4t + 2i. Consider evaluation at t + 2, i.e. the
ring homomorphism : Q[t] Q[t] sending p 7 p(t + 2), a
Solutions to Homework 11
1. Let L : K be a finite field extension, a L. We consider the K-linear map
ma : L L,
x 7 ax.
As an endomorphism of a finite-dimensional K-vector space it has a well-defined t
Partial Solutions to Homework 9
1. Prove the following isomorphism theorems:
a) Theorem: Let G be a group and H, K normal subgroups of G such that K H. Then
K is a normal subgroup of H, H/K is a norma
Partial Solutions to Homework 8
1. Stewart, part (c) for each of the exercises 12.1, 12.2, 12.3, 12.4, 12.5, 12.6.
Hint: First compute the complex zeros, and show that there exist zeros 1 , 2 such tha
Solutions to Homework 13
1. Let K be a field with 125 elements.
How many zeros in K have the polynomials t124 1, t125 1, t132 1 ?
Solution:
How many elements in Z124 have order dividing 124, 125, 132
Partial Solutions to Homework 1
The solutions to Stewart, Exercises 1.2 and 1.15 can be found on p. 190 of Stewart. (Your
answers to 1.2 should be a little more detailed.)
Exercise 1.5:
A subfield of
Partial Solutions to Homework 2
1. Stewart, exercise 1.7. This is a really long exercise. You should do it if you havent done it
in Math 113, but lets say it will not be graded.
(Besides 1.-5. on page
Dr. Martin Olbermann
Due: Wednesday, Apr. 9, in class
Math 114
Spring 2008
Homework 10
Proofs and explanations should always be written using complete English sentences. You should always explain and
Partial Solutions to Homework 3
1. a) Prove the following lemma using the quotient map r : Z[t] Zp [t].
Lemma. Let p be a prime number and f, g, h Z[t] such that f = gh. If p divides f , then
p divide
Dr. Martin Olbermann
Due: Wednesday, Feb. 27, in class
Math 114
Spring 2008
Homework 5
Proofs and explanations should always be written using complete English sentences. You should always explain and
Dr. Martin Olbermann
Due: Wednesday, Jan. 30, in class
Math 114
Spring 2008
Homework 1
Proofs and explanations should always be written, as often as possible, using complete English sentences. You sho
Dr. Martin Olbermann
Due: Wednesday, Mar. 5, in class
Math 114
Spring 2008
Homework 6
Proofs and explanations should always be written using complete English sentences. You should always explain and j
Dr. Martin Olbermann
February 20, 2008
Math 114
Spring 2008
Midterm Exam 1 - Solutions
1.
(5 points)
Let L : K be a field extension, L transcendental over K. Prove that every element
of K() which is n
Dr. Martin Olbermann
Due: Wednesday, Apr. 23, in class
Math 114
Spring 2008
Homework 11
Proofs and explanations should always be written using complete English sentences. You should always explain and
Dr. Martin Olbermann
Due: Wednesday, Feb. 27, in class
Math 114
Spring 2008
Homework 5
Proofs and explanations should always be written using complete English sentences. You should always explain and
Dr. Martin Olbermann
Due: Wednesday, Apr. 2, in class
Math 114
Spring 2008
Homework 9
Proofs and explanations should always be written using complete English sentences. You should always explain and j
Dr. Martin Olbermann
Due: Wednesday, Feb. 20, in class
Math 114
Spring 2008
Homework 4
Proofs and explanations should always be written using complete English sentences. You should always explain and
Dr. Martin Olbermann
Due: Wednesday, May 7, in class
Math 114
Spring 2008
Homework 13
Proofs and explanations should always be written using complete English sentences. You should always explain and j
Dr. Martin Olbermann
Due: Wednesday, Apr. 30, in class
Math 114
Spring 2008
Homework 12
Proofs and explanations should always be written using complete English sentences. You should always explain and
Dr. Martin Olbermann
Due: Wednesday, Mar. 19, in class
Math 114
Spring 2008
Homework 8
Proofs and explanations should always be written using complete English sentences. You should always explain and
Dr. Martin Olbermann
Due: Wednesday, Feb. 13, in class
Math 114
Spring 2008
Homework 3
Proofs and explanations should always be written using complete English sentences. You should always explain and
Dr. Martin Olbermann
Due: Wednesday, Apr. 9, in class
Math 114
Spring 2008
Homework 10
Proofs and explanations should always be written using complete English sentences. You should always explain and
Dr. Martin Olbermann
Due: Wednesday, Feb. 6, in class
Math 114
Spring 2008
Homework 2
Proofs and explanations should always be written using complete English sentences. You should always explain and j
Dr. Martin Olbermann
Due: Wednesday, Mar. 12, in class
Math 114
Spring 2008
Homework 7
Proofs and explanations should always be written using complete English sentences. You should always explain and