MATH 4321             INFORMATION SHEET FOR FINAL EXAM          SPRING 2012
                              AND COURSE SUMMARY



  * FINAL EXAM will be Tuesday May 8, 5:30-8:00 pm in PKH 103. 
  
  * All sections and questions covered on the homework website are game for
    this test.  The homework website is at
    https://vancliff.utasites.cloud/T/S12/hwk4321.html . 
  
  * My remaining office hours are: 4:00-4:50 pm on May 1 and May 3, and also 
    3:30-5:30 pm on Monday May 7.
    Ph: 817-272-3932, e-mail: vancliff@uta.edu .

  * As per the first-day handout, any student not obtaining a positive score 
    on the FINAL EXAM 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.
    
  * 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 be ready to present examples like those occurring in the homework, 
    and to show that a certain subset of a ring is a subring or an ideal 
    (1-sided or 2-sided).  You should know examples of commutative rings, 
    noncommutative rings and elements with only a left inverse but not a 
    right inverse.  You should know the Galois correspondence discussed in
    lecture on Tuesday May 1, and know how to go from the diagram of fields 
    to the diagram of groups, and vice versa.  Moreover, any question in the 
    homework is game for the test, & so is any question occurring on 
    Tests 1-4.  The ratio of old material (that tested already) to new 
    material (that not tested yet) will likely be 70:30.
    
  * As part of your study, I suggest you look over all the homework problems
    and your solutions to them, skim through the lecture notes, and look over 
    Tests 1-4 and their solutions; see  
    vancliff.utasites.cloud/T/S12/4321t1s.pdf   and 
    vancliff.utasites.cloud/T/S12/4321t2s.pdf   and
    vancliff.utasites.cloud/T/S12/4321t3s.pdf   and
    vancliff.utasites.cloud/T/S12/4321t4s.pdf 
    for the solutions to Tests 1-4 (use password  math4321  to open the 
    files).   I also suggest looking through the table of contents in your 
    book, since that gives a brief summary of the material that we have 
    covered.

  * No calculators are allowed; no notes are allowed. As per the first-day 
    handout (page 4), your cellular phone (and any other technological device)
    should be switched off and with your belongings by the wall.
    Bring some form of photo ID.
  
  * Try to keep your eyes on your own work during the 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.
  
  * If you wish to know your grade early, turn in to me a STAMPED ADDRESSED
    postcard (or a postcard inside a stamped addressed envelope) with the
    following listed on the back (I am not allowed to use e-mail to give 
    out grades): 
    (a) Math 4321 
    (b) Your full name written clearly 
    (c) Your MAV ID number 
    (d) Grade on Final = 
    (e) Grade in course = 
    I will not know your course grade until Mon May 14 at the earliest.
  
  

                       SUMMARY OF COURSE MATERIAL


1. Rings, integral domains, domains, fields, division rings (= skew fields), 
   non-commutative examples.

2. Integral domains embed in field of quotients.

3. Polynomials, factoring, homomorphisms, factor rings, 
   1st isomorphism theorem, ideals (left, right, 2-sided), 
   prime/maximal ideals and related theorems.

4. Field F inside field E (including viewpoint of E = vector space over F), 
   algebraic/transcendental extensions, finite fields.

5. Automorphisms of fields, extension theorem, splitting fields,
   separable extensions (summary of results on pg 448 of book).

6. Topics in #5 for Galois theory = correspondence between field extensions 
   and groups of automorphisms.

7. Application of Galois Theory: insolvability of the quintic ( = deg-5 poly).


#6 + #7 = the punchline for the course.