University of Massachusetts, Amherst
Spring 2018
MATH 412.1: Algebra II
(schedule number )
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Description:
This course is a continuation of Math 411. We will study properties of
rings. Rings are an abstract idea that extends the role of systems of numbers.
A ring is an algebraic system with two operations (addition and multiplication) satisfying various axioms.
Main examples are the ring of integers and the ring of polynomials in one variable. Later in the course we will apply some of the results of ring theory to construct and study fields. (A field is the simplest kind of aa ring.)
At the end we will outline the main results of Galois theory which relates properties of algebraic equations to properties of certain finite groups called Galois groups. For example, we will see that a general equation of degree 5 can not be solved in radicals.
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The required knowledge:
MAth 411.
- Some of the Basic Notions in the course:
...
The basic organizational information:
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Office :
1235I Lederle Graduate Tower.
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Email :
mirkovic@math.umass.edu
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Phone :
(413) 545-6023.
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Meet :
MW at
2:30-3:45
in Lederle Graduate Research Centert, room A301
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Office hours :
In my office: Tuesday, Wenesday 1:00-2:00.
[[Check here for changes -- temporary or permanent!]]
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Text:
Dan Saracino, "Abstract Algebra: A First Course", Second Edition, Waveland Press.
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SYLLABUS:
TO APPEAR.
Syllabus contains general information on
topics, exams, grade, course structure
and policies.
EXAM 2.
Thursday April 19th, 7-9. Place: in our classroom.
The FINAL PROJECT is
Here.
There is much work to do. The last page of the exam has detailed info on what you can and can not do.
The complete results of exams 1 and 2 should be available
Monday on moodle and in class.
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Last time we meet:
Monday May 1.
Place: in our classroom.
The exam uses the material that will be covered in the last class.
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Review session for the final exam:
Tuesday at 7.
Place: probably in our classroom.
Notes
The notes are now complete, i.e., they suffice for the Final project.
Also, typos have been fixed.
The first role of the notes is to indicate how lectures differ from the
book. Beyond this, I have tried to organize some of the material
in a more clear way.
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HOMEWORKS
There will be weekly homeworks. They will be due in class
a week after they appear on this web page, usually on Wednesday so that you
have a last chance opportunity
to ask questions on Monday.
HW2.
HW3.
HW4.
HW5.
This is also the SAMPLE EXAM 1.
HW6.
Read sections 19 and 20 in the book as well as
chapter 4 and appendix D
in the notes.
HW7.
Due in class on Wed April 11th.
Rememeber that homework 6 and redoing of the exam problems
is due Monday.
Here is
Exam 1.
HW8. This is also the Second SAMPLE EXAM. NEW!!!
Due at the exam on April 19th.
Reading assignments: these are not checked.
Optimally you should read the material before it is explained in class so that you are psychologically prepared.
EXAMS.
There will be two midterm exams and a final exam or a final project.
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EXAM 1.: Thursday March 21, 7-9 in our classroom.
(Postponed from March 7.)
The exam covers whatever we have covered in class out
of sections 16-19 in the book.
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Sample Exam 1
is the same as Homework 5.
(The actual exam will be suitable for alotted time.)
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Review Session: Monday March 5, 7-9 in our classroom.
[[We will discuss the sample exam problems and other questions
you may have.]]
In mathematics a clear explanation is called a proof
and acquiring the logical structure of proofs is one of the
basic goals of the algebra sequence.
HOW TO LEARN abstract MATHEMATICS.
The following is what I see as the {\em basic} approach
towards learning mathematics at the conceptual level.
The procedure is
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(0)
You start by hearing (or reading) of a new idea, new procedure, new trick.
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(1)
To make sense of it you check what it means in sufficiently many
examples. You discuss it with teachers and friends.
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(2)
After you see enough examples you get to the point where you
think that you more or less get it. Now you attempt
the last (and critical)
step:
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(3)
Re-tell this idea or procedure, theorem, proof or
whatever it is, to yourself in YOUR OWN words.
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More on step (3).
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Trying to memorize someone else's formulation,
is a beginning but it is far from what you really need.
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You should get to the stage where you can
tell it as a story,
as if you are teaching someone else.
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When you can do this, and your story makes sense
to you,
you are done. You own it now.
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However, if at some point you find
a piece that does not make sense, then you have to
return to one of the earlier steps (1--3) above.
Repeat this process as many times as necessary.