MATH 114: GALOIS THEORY SPRING 2008/09 PROBLEM SET 8 SOLUTIONS Let p P throughout this problem set. Let n N. We will write Fpn for the finite field of pn elements constructed in Theorem 21 in the lectures. 1. The following are to be performed only with a
MATH 114: GALOIS THEORY SPRING 2008/09 PROBLEM SET 9 SOLUTIONS As in the lecture, A B will mean `A is a subgroup of B' if A and B are both groups and `A is a subfield of B' if A and B are both fields. We will denote fields with the letters K, L, M and gro
Math 114
Professor K. Ribet
Homework assignment #15 due May 11, 2004
Let F be a nite eld; write qnfor the number of elements of F . Let n be a positive integer.
Show that the polynomial tq t F [t] factors as the product of the monic irreducible
polynomial
Garrett: Abstract Algebra
193
15. Symmetric polynomials
15.1
15.2
15.3
The theorem
First examples
A variant: discriminants
15.1 The theorem
Let Sn be the group of permutations of cfw_1, . . . , n, also called the symmetric group on n things.
For indetermi
Garrett: Abstract Algebra
203
17. Vandermonde determinants
17.1
17.2
Vandermonde determinants
Worked examples
17.1 Vandermonde determinants
A rigorous systematic evaluation of Vandermonde determinants (below) of the following identity uses the
fact that a
Garrett: Abstract Algebra
199
16. Eisensteins criterion
16.1
16.2
Eisensteins irreducibility criterion
Examples
16.1 Eisensteins irreducibility criterion
Let R be a commutative ring with 1, and suppose that R is a unique factorization domain. Let k be the
Math 114
Professor Kenneth Alan Ribet
Midterm Exam
February 24, 1992
1
Which integers d satisfy
4 points
dZ = cfw_ 51n 68m | n, m Z ?
2
20 points
3
12 points
4
9 points
Cite concrete examples of:
a. An irreducibility criterion named for a German mathemati
Math 114
Professor K. Alan Ribet
Midterm Exam
April 6, 1992
1
15 points
2
18 points
3
12 points
Cite examples of each of the following:
a. A normal extension of elds which is not separable.
b. A separable extension of elds which is not normal.
c. A K-mono
Math 114
Professor Kennneth A. Ribet
Final Exam
May 14, 1992
1
5 points
2
11 points
4 pts.
4 pts.
3 pts.
3
11 points
5 pts.
4 pts.
2 pts.
4
7 points
5
10 points
5 pts.
5 pts.
Find the sum of the roots of f (X) = X 9 5X 8 + 12X 7 6X 6 + 42.
Find the sum of
Math 114
Final Exam
Professor K. A. Ribet
May 21, 2004
This exam was a 3-hour exam. It began at 3:40PM. There were 6 problems, with all but
the second problem counting for 5 points. The second (multi-part) problem was worth 20
points. Thus the maximum pos
Math 114
Midterm Exam
Professor K. A. Ribet
Fabruary 19, 2004
This exam was an 80-minute exam. It began at 3:40PM. There were 4 problems, for which
the point counts were 6, 8, 9 and 7. The maximum possible score was 30.
Please put away all books, calculat
MATH 114: GALOIS THEORY SPRING 2008/09 PROBLEM SET 7 SOLUTIONS If K 1 and K 2 are field extensions, recall that a K-homomorphism : 1 2 is one that leaves K fixed, i.e. (a) = a for all a K. Ditto for K-isomorphism and K-automorphism. 1. Let and be algebrai
MATH 114: GALOIS THEORY SPRING 2008/09 PROBLEM SET 6 SOLUTIONS
If K is a field extension and is algebraic over K, we will define the degree of as deg fK (x), i.e. the degree of its irreducible polynomial over K. 1. Let K be a field extension. (a) Let a, b
MATH 114: GALOIS THEORY SPRING 2008/09 PROBLEM SET 5 SOLUTIONS
K will denote a field throughout this problem set.
1. Let K be a field extension and . Determine fK (x), the irreducible polynomial of over K, for the following choices of , K, . For each cas
11/11/2001 SUN
George M. Bergman
5 Evans Hall
MOFFITT LIBRARY
Spring 1994, Math 114
Final Exam
15:52 FAX 6434330
20 May, 1994
3 hours, between 4 and 8 PM
1. (40 points) Mark statements T (true) or F (false). Each correct answer will count 1
point, each in
10/05/2001 FRI 17:15 FAX 0434330 MOFFITT LIBRARY 001
George M. Bergman Spring 2001, Math 114 15 May, 2001
70 Evans Hall Final Examination 12:30-3230 PM
1. (20 points, 4 points each.) Complete each of the following denitions. (Do not give
examples or other
MATH 114: GALOIS THEORY
SPRING 2008/09
PROBLEM SET 2 SOLUTIONS
R will denote a commutative ring with unity 1 throughout this problem set. Recall that the ideal
generated by a set S R is denoted S . We will often write s1 , . . . , sn to mean cfw_s1 , . .
MATH 114: GALOIS THEORY
SPRING 2008/09
PROBLEM SET 10 SOLUTIONS
As in the lectures, we will write Gal(L/K ) for AutK (L) i K L is nite Galois.
1. Let := e2i/3 below. Determine which of the following elds K are nite Galois extensions of
Q (cf. Problems 6 a
MATH 114: GALOIS THEORY SPRING 2008/09 PROBLEM SET 1 SOLUTIONS
Throughout the problem set, unless stated otherwise, a ring will mean a commutative ring with unity 1 and will be denoted by R. We will assume that 0 = 1 in R and will write R for the set of u
MATH 114: GALOIS THEORY SPRING 2008/09 PROBLEM SET 3 SOLUTIONS D will denote an integral domain throughout this problem set. D will denote the set of units of D and 1D will denote the unity in D. Let P = cfw_2, 3, 5, 7, . . . denote the set of primes in Z
MATH 114: GALOIS THEORY SPRING 2008/09 PROBLEM SET 4 SOLUTIONS
D will denote a unique factorization domain throughout this problem set. For n N, n 2, Zn will denote the ring of intergers modulo n. We will denote the elements of Zn as 0, 1, . . . , n 1 ins
Math 114
Midterm Exam
Professor K. A. Ribet
April 8, 2004
This exam was an 80-minute exam. It began at 3:40PM. There were 4 problems, for which
the point counts were 7, 8, 8 and 7. The maximum possible score was 30.
Please put away all books, calculators,