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,
PROBLEM SET # 9
MATH 114
Due April 6.
1. Let n = p, or 2p where p is a prime number. Prove that the Galois group of the
polynomial xn 1 over any field F is cyclic.
2. Show that the Galois group of x15 1 over Q is isomorphic to Z2 Z4 .
By Fq we denote the
PROBLEM SET # 12
MATH 114
Due April 27.
1. Trisect the angle of 18 by ruler and compass.
2. Let l be the least common multiple of m and n. Assume that regular m-gon
and regular n-gon are constructible by ruler and compass. Prove that regular l-gon
is also
PROBLEM SET # 13
MATH 114
Due May 4.
1. Prove that the Galois group of f(x) = x5 + x4 4x3 3x2 + 3x + 1 over Q
is cyclic of order 5. Hint: let be 11-th root of 1. Prove that f(x) is the minimal
polynomial for + 1 .
2. Let p be an odd prime, be a primitive
PROBLEM SET # 11
MATH 114
Due April 20.
1. Let E and B be normal extensions of F and E B = F . Prove that
AutF EB
= AutF E AutF B.
2. Find the Galois group of the polynomial (x3 3) (x3 2) over Q.
3. Let f (x) be an irreducible polynomial of degree 7 solv
PROBLEM SET # 3
MATH 114
Due February 9.
1. Read notes on Sylow theorems. Prove the last corollary in these notes.
2. If p is prime and p divides |G|, then G has an element of order p.
3. Let p and q be prime and q 6 1 mod p. If |G| = pn q, then G is solv
PROBLEM SET # 5
MATH 114
Due April 13.
1. Find the Galois group of the polynomial x4 + 8x + 12 over Q.
2. Find the Galois group of the polynomial x4 + 3x + 3 over Q.
3. Find the Galois group of x6 3x2 + 1 over Q .
4. Assume that a polynomial x4 + ax2 + b
PROBLEM SET # 4
MATH 114
Due February 16.
Read chapter 1 in Artins book. We assume that all fields are commutative in
homework problems.
1. Let F be a field, and F [i] denote the set of all expressions a + bi, with a, b F .
Define addition and multiplicat
PROBLEM SET # 6
MATH 114
Due March 2.
1. Let
F be a field. Prove that every extension F E of degree 2 is isomorphic
d for some d F which is not a perfect square in F .
to F
2. Find the degree of the splitting field of the polynomial x3 + x + 1 Q [x]. Hin
SOLUTIONS FOR REVIEW EXERCISES
MATH 114
1. Let G be a transitive subgroup of Sn .
(a) Prove that if n is prime, then G contains an n-cycle.
(b) Show that (a) is not true if n is not prime.
Solution. The number of elements in an orbit divides the order of
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
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
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
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
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
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
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
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
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
PROBLEM SET # 1
MATH 114
Due January 26.
1. Prove that a subgroup of a cyclic group is cyclic. (You have to consider both
infinite and finite cyclic group).
2. Let G be a group and the order of G be even. Show that there is a G of order
2. Hint if a2 6= 1
PROBLEM SET # 2
MATH 114
Due February 2.
1. An automorphism of a group G is an isomorphism from G to itself. Denote by
Aut G the set of all automorphisms of G.
(a) Prove that Aut G is a group with respect to the operation of composition.
(b) Let G be a fi
PROBLEM SET # 7
MATH 114
Due March 16.
1. Find the Galois groups of the following polynomials over Q:
(a)x4 + x2 + 1;
(b)(x2 2)(x2 3)(x2 5);
(c)x6 3;
(d)x5 2.
2. Which of the following are normal extensions?
(a)Q Q[x]/(x3 + x + 1);
(b)Z2 Z2 [x]/(x3 + x +
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 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 , . .
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
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