Math 547, Exam 2. 3/19/10.
Name:
Read problems carefully. Show all work.
No notes, calculator, or text.
The exam is approximately 15 percent of the total grade.
There are 100 points total. Partial credit may be given.
1. (15 points)
(a) (5 points) Giv
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Quiz for March 4, 2005
1. Let F E be elds with E a nite dimensional vector space over F . Let R
be a ring with F R E . Prove that R is a eld.
ANSWER: Take u R , with u = 0 . I will prove that F [u] is a eld. This is
enough because, once I
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Quiz for April 18, 2005
1. Let K C be elds, and let f (x) be an irreducible polynomial in K [x] .
Prove that f (x) has DISTINCT roots in C .
ANSWER: This is a proof by contradiction. Suppose C is a root of f (x) of
multiplicity at least 2
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Quiz for April 4, 2005
2i
1. Let = e 17 and let K be the eld Q[ ] . Find an element u2 in K with
dimQ Q[u2 ] = 4 .
ANSWER: I use Galois Theory. If AutQ K , then ( ) must be a root of the
minimal polynomial of . So ( ) must equal j for som
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Quiz for March 28, 2005
2i
1. Let = e 7 and let K be the eld Q[ ] . Find an element u1 in K with
dimQ Q[u1 ] = 2 .
ANSWER: Let u1 = + 2 + 4 . Observe that
u2 = 2 + 2 3 + 2 5 + 4 + 2 6 + .
1
It follows that
u2 + u1 = 2( + 2 + 3 + 4 + 5 + 6
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Quiz for March 16, 2005
1. Prove that a regular 7 -gon is not constructible.
2i
ANSWER: A regular 7 -gon is constructible if and only if = e 7 is
constructible. It is clear that is a root of x7 1 . It is also clear that
x7 1 = (x 1)g (x)
Review sheet for Exam 4
1. Be able to do all of the assigned Homework problems from March 23 and April
4.
2. Find the lattice of subelds between Q and F , where F is the splitting eld
of x3 2 over Q .
3. Find the lattice of subelds between Q and F , where
Math 547, Exam 4, Spring , 2005
The exam is worth 50 points. Each problem is worth 10 points.
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Take enough space for each problem. Turn in
your
Math 547, Final Exam, Spring , 2005
The exam is worth 100 points. Each problem is worth 11 1/9 points.
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Take enough space for each problem. Tur
The answer is correct!
Let u =
3
2 . In class we calculated that the inverse of 1 + 2u + 3u2 in Q[u] is
1
(9 + (27u + 450)(3u 2).
801(9)
Of course, this answer may be cleaned up to become:
1
(81u2 + 1296u 891).
801(9)
We now check our answer. We notice th
Math 547, Exam 4, Spring , 2005 Solutions
The exam is worth 50 points. Each problem is worth 10 points.
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Take enough space for each problem. Tu
PRINT Your Name:
Quiz for January 21, 2005
Prove that the composition of two ring homomorphisms is a ring homomorphism.
ANSWER: Suppose : R S and : S T are ring homomorphisms.
We see that
( )(1) = (1)
by the denition of composition;
(1) = (1)
because is
Homework 8 Solutions.
6.4 #1. Determine the splitting elds in C for the following polynomials (over Q).
(a) x2 2. The roots are cfw_ 2; hence, a splitting eld is Q( 2).
(b) x2 + 3. The roots are cfw_ 3; hence, a splitting eld is Q( 3).
(c) x4 + x2 6. Sinc
Math 547, Exam 3 Information.
4/16/10, LC 303B, 10:10 - 11:00.
Exam 3 will be based on:
Homework and textbook sections covered by lectures 3/15 - 4/12.
(see http:/www.math.sc.edu/boylan/SCCourses/547Sp10/547.html)
At minimum, you need to understand how to
Math 547, Exam 2 Information.
3/19/10, LC 303B, 10:10 - 11:00.
Exam 2 will be based on:
Homework and textbook sections covered by lectures 2/3 - 3/5.
(see http:/www.math.sc.edu/boylan/SCCourses/547Sp10/547.html)
At minimum, you need to understand how to d
Math 547, Exam 1 Information.
2/10/10, LC 303B, 10:10 - 11:00.
Exam 1 will be based on:
Sections 5.1, 5.2, 5.3, 9.1;
The corresponding assigned homework problems
(see http:/www.math.sc.edu/boylan/SCCourses/547Sp10/547.html)
At minimum, you need to under
Math 547, Exam 1. 2/9/10.
Name:
Read problems carefully. Show all work.
No notes, calculator, or text.
The exam is approximately 15 percent of the total grade.
There are 100 points total. Partial credit may be given.
1. (10 points) Let R be a commutat
Math 547, Exam 3. 4/16/10.
Name:
Read problems carefully. Show all work.
No notes, calculator, or text.
The exam is approximately 15 percent of the total grade.
There are 100 points total. Partial credit may be given.
1. (20 points) Let F and K be eld
Homework 7 Solutions.
6.1 #12. Assuming that is transcendental over Q, show that either + e or e is transcendental
over Q.
Proof. Suppose, by way of contradiction that
+ e is algebraic over Q with degree m, and that
e is algebraic over Q with degree n.
Homework 6 Solutions.
6.1 #1 c. Show that
3 + 5 is algebraic over Q.
Proof. Compute as follows:
a = 3 + 5 a2 = 3 + 2 15 + 5 = 8 + 2 15 (a2 8)2 = 4 15 = 60
a4 16a2 + 4 = 0.
Hence, for f (x) = x4 16x2 + 4 Q[x], we have f (a) = 0, so a is algebraic over Q.
Homework 5 Solutions.
4.2 #1 d. Use the division algorithm to nd the quotient and remainder when f (x) = 2x4 + x3 6x2
x + 2 is divided by g (x) = 2x2 5 over Q.
Solution: Long division gives:
1
1
f (x) = g (x) x2 + x
2
2
3
1
+ x .
2
2
4.2 #2 b. Use the d
Math 547, Final Exam, Spring , 2005
The exam is worth 100 points. Each problem is worth 11 1/9 points.
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Take enough space for each problem. Tur