(last revision 4/24/14, 10:00 pm)
MATH 4321 INFORMATION SHEET FOR FINAL TEST SPRING 2014
(scroll down to see a summary of the course material)
The Final Test will be 2:00-4:30 pm on Tues May 6 in PKH 302; it is worth 40% of your course grade.
The relevant sections for the Final Test are all the sections covered on the homework website throughout the semester.
My office hours are Tues and Thurs 3:30-4:10 pm through Thurs May 1 and the GTA's office hours are given on my course website. In addition, I will have office hours 3:30-5:30 pm on Monday May 5.
As per the course information sheet, any student not obtaining a positive score on the Final Test for this class will not pass this class.
You should learn definitions such as ring, subring, division ring, field, unit, zero divisor, integral domain, homomorphism, isomorphism, characteristic, automorphism, field of quotients/fractions, (ir)reducible polynomial, (left/right) ideal, factor ring, kernel, coset, maximal ideal, prime ideal, prime field, linear (in)dependence, dimension/degree, basis, algebraic/transcendental element/extension, finite extension, algebraic closure, algebraically closed, (primitive) n'th root of unity, normal extension, separable extension, perfect field, Galois field/group, etc.
You should be ready to present examples like those occurring in the homework, and be able to show that a subset of a ring is a subring or left/right ideal. You should know examples of commutative rings, noncommutative rings and elements with only a left inverse but not a right inverse (or vice versa). You should know the statement of any theorem with a title, such as the first isomorphism theorem (pg 242) & the theorems summarized on page 248 (in the box) and how to use such a theorem (but not its proof). You should know the Galois correspondence discussed in lecture on Thursday April 24, and know how to go from the diagram of fields to the diagram of groups, and vice versa, as demonstrated in the examples on the handout of Thursday April 24.
Any question in the homework or the Quiz or Tests 1-3 is game for the test. I recommend you look over the Quiz and Tests 1-3; the solutions to those are posted at my website. The ratio of old material (that tested already) to new material (that not tested yet) will likely be 70:30.
No calculators are allowed; no notes are allowed; no phones, no tablets. Your belongings will need to be by the wall as in the previous tests.
Bring some form of photo ID.
Try to keep your eyes on your own work during the Final Test.
If you wish to leave the room during the test, you should ask permission first & turn in your test to me. Only in exceptional circumstances will I let you continue the test should you return. (So it is better to be 3 minutes late to the test, rather than ask to go to the bathroom during the test.) If you finish early but prefer to stay in the room, then you should NOT get out any work, book nor item, no matter what the subject matter is, without first checking with me. Should you wish to leave the test early, then you may.
It is your responsibility to be on time.
See below for a summary of the course material.
Rings, integral domains, domains, fields, division rings (= skew fields), non-commutative examples.
Integral domains embed in field of quotients.
Polynomials, factoring, homomorphisms, factor rings, 1st isomorphism theorem, ideals (left, right, 2-sided), prime/maximal ideals and related theorems.
Field F inside field E (including viewpoint of E = vector space over F), algebraic/transcendental extensions, finite fields.
Automorphisms of fields, extension theorem, splitting fields, separable extensions (summary of results on pg 448 of book).
Topics in #5 for Galois theory = correspondence between field extensions and groups of automorphisms.
Application of Galois Theory: insolvability of the quintic ( = deg-5 poly).
#6 + #7 = the punchline for the course.