M a 5c HOMEWORK 1 SOLUTION
SPRING 09
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote, unless stated otherwise. Page 519,4. Dene : Q[ 2] Q[ 2] a + b 2 a b 2, via where 2) 2 2 a, b Q. Then (a + b 2) + (c + d = a b
M a 5c HOMEWORK 5 SOLUTION
SPRING 09
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote, unless stated otherwise. Page 595,4. Suppose f (x) = (x i ) L[x], and f (x) = m fj (x) j =1 with fj (x) irreducible over K .
Math 5c, Spring 10 Problem Set 5 Due May 4, 4 pm All numbered problems are from Dummit and Foote, Third Ed. Key Concepts: Galois groups of Composite elds and Cyclotomic extensions
14.4 4, 5 14.5 7, 13 14.6 20
1
Math 5c, Spring 10 Problem Set 1 Due Tuesday, April 4pm All numbered problems are from Dummit and Foote, Third Ed. Key Concepts: Field extensions, Algebraic closures, embeddings over a eld.
13.1 4 13.2 12, 13 1. Let and be two algebraic elements over a el
Math 5c, Spring 10 Problem Set 2 Due Tuesday, April 13 4pm All numbered problems are from Dummit and Foote, Third Ed. Key Concepts: Splitting Field, Normal eld, Seperable and inseperable extensions
13.4 5 13.5 11 1. Let f (x) F [x] be a polynomial of degr
Math 5c, Spring 10 Problem Set 2 Due Tuesday, April 20 4pm All numbered problems are from Dummit and Foote, Third Ed. Key Concepts: Galois extensions
14.1 4, 10 14.2 4, 9, 16
1
Math 5c, Spring 10 Problem Set 4 These problems are not due, but the material covered on them will be considered fair game for the midterm and nal exam. All numbered problems are from Dummit and Foote, Third Ed. Key Concepts: Galois extensions
14.2 7, 29
Math5c, Spring 10 Problem Set 6 Due May 11, 4pm All numbered problems are from Dummit and Foote, Third Ed. Key Concepts: Galois groups of polynomials, Krummer extensions
14.6 16, 19, 38, 47 14.7 4
1
Math 5c, Spring 10 Problem Set 6 Due May 18, 4 pm All numbered problems are from Dummit and Foote, Third Ed. Key Concepts: Krummer extensions, Norms, traces, representations
14.2 17cd Let be a primitive nth root of unity. Let K = Q( ). 1. If n = pr with r
Math 5c, Spring 10 Problem Set 6 Due May 18, 4 pm All numbered problems are from Dummit and Foote, Third Ed. Key Concepts: Representations, canonical forms of matrices
18.1 3, 4, 6abc, 10, 12.2 1
1
Math 5c, Spring 10 Problem Set 9 Due May 3, 4 pm All numbered problems are from Dummit and Foote, Third Ed. Key Concepts: Shurs Lemma, Characters
18.1 9, 17, 19 18.3 4, 10
1
M a 5c HOMEWORK 2 SOLUTION
SPRING 09
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote. Page 545,5. Suppose K is a splitting eld over F for a collection of cfw_fi (x). Let f be an irreducible polynomial over F wit
M a 5c HOMEWORK 3 SOLUTION
SPRING 09
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote, unless stated otherwise. Page 567,4. Suppose there is an isomorphism : Q( 2) Q( 3). Since xes Q and is a homomorphism, we mus
Math 5c Midterm Due Wednesday, April 28, 4 pm Do the following 5 problems in 3 hours. You may cite any theorems proven in class or Dummit and Foote, but prove any material you use from other sources. You may use technology for calculations. 1. Let E and K
M a 5c HOMEWORK 6 SOLUTION
SPRING 09
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote, unless stated otherwise. Page 618,16. The polynomial is irreducible by brute force. The resolvent cubic is x3 16x2 + 48x + 64
M a 5c HOMEWORK 7 SOLUTION
SPRING 09
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote, unless stated otherwise. Page 582,17. c) Gal( ) = cfw_1, , with (a + b D) = a b D. K/F Therefore NK/F (a + b D) = (a + b D)(a
M a 5c HOMEWORK 8 SOLUTION
SPRING 09
The exercises are taken from the text, Abstract Algebra (third edition) by Dummit and Foote, unless stated otherwise. Page 852,3. Let : G F be a degree 1 representation of G. Since im is a subgroup of an abelian group