Ring theory: factor rings, Euclidean domains, polynomials over fields and over the integers, maximal ideals in rings, quotient fields.
Field extensions: simple extensions, algebraic and transcendental numbers, the degree of an extension, normality and separability, finite fields.
The Galois group: Normal closure, the Galois correspondence, soluble and simple groups, ruler and compass constructions, solution of equations by radicals.
Additional topics:transcendental degree, the general polynomial equation, the general Galois group, calculating the Galois group, the regular $n$-gon, quadratic and cyclotomic fields, the fundamental theorem of algebra.
Course details
This class will meet 28 times, and we will cover more or less the entire textbook. Additionally, the expectation is that, in addition to the weekly written homework, you will read every word of the textbook and additional notes.
Grading policy
The midterm dates below are tenative (and may be adjusted if the pace of the course is adjusted), but the date of the final exam is set in stone; make your summer travel plans accordingly. If you have a conflict with the final exam (e.g., another final) please let me know ASAP.
Homework
40%
(Bi-weekly, due Thursdays, 5pm)
Midterm
30%
(Th, March 26)
Final Exam
30%
(W May 6, 8-10:30am, W304)
Calculators, notes, and textbooks are not allowed in exams or quizzes.
The final letter grades will be curved, but the following table gives a lower bound on your grade:
85%
A
70%
B
55%
C
Homework
There will be homework assigned roughly every week, due on every other Tuesday at 5pm (in my mailbox). There
will be many simple problems, checking your understanding of the
definitions, that will be collected and graded for completness but not
correctness. Most weeks there will be a number of proofs assigned. You
are expected to write them up very carefully.
3-6 of problems will be carefully graded, and you will receive an additional 20 for completing the assignment.
Homework assignments will typically be worth 100 points (20 for completeness, and 80 for graded problems).
The homework assignments are available at this link, and will be updated after each lecture.
Plagarism Policy
Remember that copying another student's work is a violation of the
Honor Code and will be treated as such. If you must leave class during
an exam for any reason, please leave all of your belongings
(including your handheld supercomputer phone!).
For homework: you are free to consult any sources (animate or
inanimate) while doing your homework (working in groups is
encouraged!), but if you use anything (or anyone) other than your
class notes or the texts listed above, you should say so on your
homework -- please state at the end of every problem any sources used.
On the other hand, you are expected to make an honest attempt to do every problem on your own before consulting other sources. Remember that copying another student's work is a violation of the Honor Code and will be treated as such.
A good rule of thumb to avoid plagarism is the following -- when doing
the final write up of a problem, do not have any text books, web
pages, or classmate's write up in front of you. If you get stuck when
writing up an assignment, go back and look again; just make sure that
you organize the mathematics in your head before writing a proof
rather than copying a solution from some source. This is a
generous homework policy. Please do not abuse it.
Overloads
Ken Mandelberg handles all overloads for the department.
To request an overload you must complete the form here and have it signed by your PACE or major advisor.