MATH 237
FALL 2006
SOLUTIONS FOR SECOND MIDTERM EXAM
NOVEMBER 29, 2006
1. (10 points) Let F be a nite eld with pn elements where p is a prime.
Show that if F generates the cyclic group F , then the degree of
over Zp is n.
Solution: Since generates the cy
1. MATH 237 Sample Problems for first midterm exam
On this sheet R and S always denote rings and F and E always denote elds. Further F E
means that F is a subeld of E .
(1) Show that the characteristic of an integral domain must be 0 or a prime number.
(2
MATH 237
FALL 2006
SOLUTIONS FOR FIRST MIDTERM EXAM
OCTOBER 23, 2006
1. Give denitions of each of the following. (4 points each)
(a) Principal ideal domain Solution: See 8.6.2 in the textbook.
(a) Integral domain Solution: See 6.2.8 in the textbook.
(c) M
Math 237
First midterm exam
October 20, 2008
Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem.
Show all of your work.
1. Give denitions of each of the following. (4 points each)
(a) Principal ideal domain
Soluti
Math 237
First midterm exam
October 22, 2012
Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem.
Show all of your work.
1. Find the minimal polynomial f (x) for =
(5 points each)
2+
3 over each of the following el
Math 237
First midterm exam
October 20, 2010
Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem.
Show all of your work.
1. Give denitions of each of the following. (4 points each)
(a) Principal ideal domain
Soluti
Math 237
First midterm exam
October 21, 2013
Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem.
Show all of your work.
1. (20 points) Find the minimal polynomial f (x) for =
need not prove that your monic polynom
Math 237
Second midterm exam
November 24, 2008
Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem.
Show all of your work.
1. Give denitions of each of the following. (5 points each)
(a) Galois group
Solution: See
Math 237
Second midterm exam
November 26, 2012
Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem.
Show all of your work.
Note that there are ve problems, the fth one being on the back of this page.
1. Give deniti
Math 237
Second midterm exam
November 22, 2010
Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem.
Show all of your work.
Note that there are ve problems, the fth one being on the back of his page.
1. Give denitio
Math 237
Second midterm exam
November 20, 2013
Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem.
Show all of your work. Note that there are problems on both sides of this sheet.
1. Let = e2i/7 = cos(2/7) + i sin
MATH 237
FALL 2006
FINAL EXAM ANSWER SHEET
DECEMBER 19, 2006
1. (20 points) Let K/F be a Galois extension with Gal(K/F ) = G. Let
H1 and H2 be subgroups of G and let L1 and L2 be the corresponding
intermediate elds. Show that the intermediate eld correspo
Math 237
Final exam
December 10, 2008
Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem.
Show all of your work.
This is a takehome exam. It is due at 2:00 pm on Tuesday, December 16, in my oce, Hylan
819, or in m
Math 237
Final exam
December 12, 2012
Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem.
Show all of your work.
This is a take home exam. It is due at 2:00 pm on Monday, December 17, in my oce, Hylan
819, or in m