Math 123: Study Guide for Midterm 1
Here is a list of the sections from the book that you need to know for the
Midterm.
Chapter 13. Sections:
1, only on p. 492.
2,3 all.
4 without the Constructions
5,6 all.
7,8 youre not responsible for these, even f
Kummer Theory
April 13, 2010
1
HOMEWORK DUE APRIL 22:
Exercise 1 Please hand in (to me in Tuesdays class) correct answers to all the problems you got
wrong in the April 8 exam.
Exercise 2
1. Let L/K be a Galois extension with Galois group G = Gal(L/K). Le
Math 123: Practice Midterm 2
Note: This practice exam is strictly longer and harder than the actual
one.
Problem 1. Let R be a Euclidean domain, and let M be a nitely
generated module over R. For each of the following statements, determine
whether it is n
Intro to the Fundamental Theorem of Galois Theory
March 30, 2010
ANNOUNCEMENT: New date for next midterm: April 8.
Reading: [Artin] pp. 493-508, 537-547
1
Alternate characterizations of Galois extensions
Here is a repeat, in somewhat more extended form, o
Math 123: Practice Midterm 1
Note: This practice exam is strictly longer and harder than the actual
one.
Problem 1. Give an example of a polynomial f (x) Q[x] such that
the splitting eld L of f (x) over Q has degree equal to the degree deg f, but
such tha
An example of a universal Problem and its universal solution
1
The Universal solution to the problem of inverting a given
set of elements in a ring.
Exercise 1 The set of all rational numbers of the form a/2n for some integer a and some natural
number n s
Math 123
Problem Set 2
1. Artin, page 442, exercise #9
(a) Recall that the norm of a number in Z[ 5] is given by N (a+b 5) = a2 +5b2 .
Also, Artin proves that this norm is multiplicative.
Now, N (2) = 4, and since the norm of an element in Z[ 5] cant be 2
Note: These solutions are slightly more detailed than the ones you
had to put on the exam.
Problem 1.
Let F27 be the nite eld with 27 elements. Does there exist F27
such that 2 = 1?
Solution 1. Suppose such an F27 exists. Since the polynomial
x + 1 F3 [x]
Math 123 Problem Set 1
Exercise 1
Let p, q denote complex numbers. Consider the equation
y 3 + py + q = 0
We want to nd the roots of y 3 + py + q. Introduce a new symbol z by putting
y := z
p
3z
Now show that if y is a root of the displayed cubic equatio
Problem 1.
a) No. cfw_2 is a lineraly independent set in Z (considered as a free
Z-module), but cannot be extended to a basis.
b) No. cfw_2, 3 is a spanning set of Z considered as a free Z-module),
because 3 2 = 1, but does not a contain a basis of Z.
Eac
Math 123
Introduction
1. A very very brief description of the large themes of the course.
Choose a some integer and consider the polynomial
X5 X + a
(e.g., for a = 1, 1, 2, 2, . . .) and ask for its roots. It still fascinates me that (except for very
rare