Abstract Algebra II
 
Galois Theory, Spring 2018  UNT


Professor: Dr. Anne Shepler
Lecture:
Audb 212, Tues/Thur 11--12:20 pm
Office:  GAB 471B,   phone:  940-565-4943
Office hours: Tues 3--5pm and Thurs 2--4; by appt
Web: http://www.math.unt.edu/~ashepler/Math4510S18.html
Prerequisites: Math 3510 with grade of "C" or better
Text:  A first course in abstract algebra by John B. Fraleigh (7th Edition)

Course Description:  This course introduces Galois theory, one of the most fruitful fields of mathematics.  Galois theory addresses the problem of finding solutions to polynomial equations.  It uses group theory to analyze roots of polynomials.  We will cover field extensions, seperable extensions, normal field extensions, splitting fields, ruler and compass constructions, applications to classical Greek geometry, and solvability by radicals of cubic and quartic equations.  We will also introduce theorems like the Fundamental Theorem of Symmetric Polynomials, the Fundamental Theorem of Algebra, the Theory of the Primitive Element, and the Fundamental Theorem of Galois Theory.

Grading:  Course grade is based on

Date/Time of Final:   Tuesday, May 8, 2018 at 10:30-12:30 in our regular classroom.

Exams and Final:  No hats or smart devices, including watches. You must take the final exam to pass the course.   You  MUST  take the final exam at its scheduled time and place and also take the midterms on the scheduled dates at the scheduled times.  There will be NO make-up exams.  Plan your schedule accordingly.  In the event of a documentable emergency or illness, contact the professor immediately (BEFORE the scheduled exam when possible).  If everyone does well, everyone will receive a good grade, so study together and avoid competition.  Count your points on exams and homework to be sure the totals are correct and keep a record of all your scores.

Homework:   Come to lecture with your homework stapled and ready to turn in at the beginning of class.  Your lowest homework score will be dropped automatically to cover homework missed due to illness, family emergency, transportation problems, oversleeping, injury, work schedule, completing the wrong section, completing the wrong problems, tornados, eclipses, etc.   Homework that is difficult to read will earn a zero score.  Only hardcopy written work will be accepted, no faxes or emails or scans will be accepted.  (ODA students may have exceptions.)

Disabilities: It is the responsibility of students with certified disabilities to provide the instructor with appropriate documentation from the Office of Disability Accommodation before the third week of class.



Extra Credit:  Do not expect to be able to do some extra work to help your grade either before or after the final exam. There will be no extra credit during the semester, except possibly an extra problem on an exam. You must complete the assigned work on time.

Cheating:  Academic honesty is a minimal expectation for this and all UNT classes.  Anyone caught cheating will receive an F for the course.  Furthermore, a letter will be sent to the appropriate dean who may take further disciplinary actionAny gadget out (except an approved calculator) during an exam will be interpreted as academic dishonesty.  (Keep your electronic gizmos zipped in a backback.)

Email:  Instructors sometime receive 50-100 academic emails a day, and UNT sometimes moves faculty emails.  Help your emails get prompt attention.  Put all your information into ONE email and keep the email BRIEF.   Write in complete sentences with appropriate grammar. Your subject line should include course, name, and topic, for example,"SUBJECT: Math 4510, Jane Doe, brain surgery".   Save your math questions for in person;  it is too difficult to explain math over email because the limitions in notation often cause confusion.

Office Hours:  Outside of office hours, your instructor is likely advising graduate students, refereeing papers, writing grant proposals, preparing manuscripts for publication, doing editorial work for journals, reviewing budgets, writing talks, skyping or phoning with collaborators, reviewing university programs, writing reports, designing new courses, grading papers, evaluating grant applications, preparing lectures and exams, organizing conferences, reviewing graduate and job applications, revising articles, writing letters of recommendation, reviewing PhD theses, proof reading manuscripts, attending department meetings, updating webpages, writing computer code, completing mathematical computations, and proving new theorems.  Instead of just dropping by, it is best to make an appointment if you can not make office hours.

Written work: Show all your work (in clear steps) on exams and homework.  No (or little) work shown usually earns no credit---even if the answer is correct.   Your proofs (and solutions) will be graded on four "C's":  solutions must be clear, concise, complete, and correct.   Your audience should be an average student in this course, someone who has read the problem but does not know a solution. Rule of thumb:  If a fact is "obvious", then it can be proved in one or two lines, so you might as well include those lines.  The back of the book contains hints, not solutions, to problems: your solution must contain more detail than in the back of the book or any solution guide.  Copying the hint from the back of the book will earn little or no credit.  In general, proofs without enough detail or with confused steps will earn little or no credit. 

Expectations:  You are expected to come to every lecture and come on time.  Plan ahead so you are not late.  You are responsible for everything that happens in class.   You are expected to read the assigned sections and work on the homework problems immediately after they are assigned. You should be prepared to ask questions, take notes, and look alive in class.  Leave all electronic gadgets turned off and out of reach.  Feel free to bring beverages to class (coffee, cola, tea, water, etc.) or quiet snacks to help you participate. It is the student's responsibility to obtain notes from another student if class is missed.

Learning Outcomes:   Students will obtain computational and theoretical experience with the Galois correspondence and use group theory to analyze roots of polynomials.  Students will be able to work with field extensions, compute Galois groups for polynomials, and illustrate the Galois correspondence using lattice diagrams.  Proof writing skills should also improve over the course.

Rough Schedule  (subject to change):   Weeks 1--4: rings of polynomials, factorization of polynomials, homomorphisms and factor rings, prime and maximal ideals; Weeks 5--6: extension fields, vector spaces, algebraic extensions, geometric constructions;  Weeks: 7--9: finite fields, automorphisms of fields, isomorphically extending fields;  Weeks 10--11: splitting fields, separable extensions; Weeks 12-14:  Galois theory and applications.









Abstract Algebra II
Homework Problems

SECTION
PROBLEMS
DUE