MATH 4321          INFORMATION SHEET FOR FINAL EXAM             SPRING 2011
                            AND COURSE SUMMARY


  * FINAL EXAM will be Fri May 13, 11:00 am - 1:30 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/S11/hwk4321.html . 
  
  * My office hour this week is 1:30-3:20 pm on Thurs May 12 in PKH 462.
    Ph: 817-272-3932, e-mail: vancliff@uta.edu .
  
  * You should learn definitions such as ring, subring, division ring, field,
    (integral) domain, unit, homomorphism, isomorphism, automorphism, 
    zero-divisor, characteristic, zero of a polynomial, field of fractions,
    quaternions, left/right ideal, 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, (primitive) n'th root of unity, normal extension, 
    seperable extension, perfect field, Galois field/group, algebraically 
    closed. 
    
  * 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 3, 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 per the first-day handout, any student not obtaining at least 25% on 
    the Final Exam for this class will not pass this class.
  
  * Only calculators specified on the first-day handout will be permitted in 
    the test.  If you bring one, make sure the batteries are working. Also 
    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 16 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.