Algebra - II
MATH 5623      Spring 2008

Course Syllabus: [pdf]
Exam dates:
     1st Midterm: February 15, Friday, 3:30 p.m. to 5:30 p.m.
     2nd Midterm: April 4, Friday, 3:30 p.m. to 5:30 p.m.
     Final Exam : May 2, Friday, 8:00 a.m. to 9:50 a.m.
    Link to the homepage of Algebra-I: MATH 5613

Weekly Syllabus:

1. (Week of Jan 7)
     Jan 7: Distribute syllabus; review some module theory from Algebra-I.
     Jan 9: Projective modules.
     Jan 11: Exact functors.
     Homework 1: [pdf]    (Due Jan 18, Friday.)    [Solutions]
2. (Week of Jan 14)
     Jan 14: Injective modules.
     Jan 16: Projectives and Injectives, cotd.
     Jan 18: Hom and Duality.
     Homework 2: [pdf]    (Due Jan 23.)    [Solutions]   [Solution to the bonus exercise]
3. (Week of Jan 21)
     Jan 21: MLK day.
     Jan 23: Tensor products; fundamental isomorphisms.
     Jan 25: Adjoint functors; Flat modules.
     Homework 3: [pdf]    (Due Jan 30.)    [Solutions]
4. (Week of Jan 28)
     Jan 28: Bilinear forms.
     Jan 30: Symmetric and alternating bilinear forms.
     Feb 1: Multilinear algebra.
     Homework 4: [pdf]    (Due Feb 6.)    [Solutions]
5. (Week of Feb 4)
     Feb 4: Modules over PIDs.
     Feb 6: Modules over PIDs, cotd.
     Feb 8: Modules over PIDs, cotd.
     Homework 5: [pdf]    (Due Feb 13.)    [Solutions]
6. (Week of Feb 11)
     Feb 11: Modules over PIDs.
     Feb 13: Review of the theory of modules.
     Feb 15: Office hour. (Midterm exam from 3:30 to 5:30 p.m.)
     There will be no homework for this week.
7. (Week of Feb 18)
     Feb 18: Extension of fields.
     Feb 20: Extension of fields, cotd.
     Feb 22: Extension of fields, cotd.
     Homework 6: Sec V.1 of Hungerford's Algebra: 6, 7, 8, 10, 12, 14, 15. (Due Feb 27.)    [Solutions]
8. (Week of Feb 25)
     Feb 25: Extension of fields.
     Feb 27: Automorphisms, fixed field, Galois extensions.
     Feb 29: Fundamental theorem of Galois theory.
     Homework 7: Sec V.1: 18, 19, 20, 22; Sec V.2: 2, 4. (Due Mar 5.)    [Solutions]
9. (Week of Mar 3)
     Mar 3: Proof of the fundamental theorem.
     Mar 5: Proof, cotd.
     Mar 7: Splitting fields.
     Homework 8: Sec V.2: 11, 12, 15; Sec V.3: 5, 8, 9. (Due Mar 14, Friday.)    [Solutions]
10. (Week of Mar 10)
     Mar 10: Splitting fields and algebraic closure.
     Mar 12: Separable extensions.
     Mar 14: Normal extensions.
     Homework 9: Sec V.3: 12, 13, 14, 17, 18, 22. (Due Mar 26.)    [Solutions]
11. (Week of Mar 17) *****SPRING BREAK*****
12. (Week of Mar 24)
     Mar 24: Normal extensions; Normal closure.
     Mar 26: Fundamental theorem of algebra.
     Mar 28: Galois group of a degree 3 polynomial.
     Homework 10: Sec V.4: 2, 3, 5, 8, 9, 11. (Due Apr 2.)    [Solutions]
13. (Week of Mar 31)
     Mar 31: Galois group of quartic and other polynomials.
     Apr 2: Review of Galois theory.
     Apr 4: Office hour. (Midterm exam from 3:30 to 5:30 p.m.)
     Practice questions for the midterm exam:
     Sec V.1: 1, 2, 3, 5--10, 11a, 15, 19--23;   Sec V.2: 2, 4, 6, 11, 12, 14, 15;
     Sec V.3: 5--15, 17, 18, 20--24;   Sec V.4: 1--3, 5, 6, 8--12.
     There will be no homework for this week.
14. (Week of Apr 7)
     Apr 7: Finite fields.
     Apr 9: Finite fields.
     Apr 11: Purely inseparable extensions.
     Homework 11: Sec V.5: 3, 7, 9; Sec V.6: 4, 7, 8, 13; Sec V.7: 2, 5, 8. (Due Apr 23.)    [Solutions]
15. (Week of Apr 14)
     Apr 14: Separable closure.
     Apr 16: Norm, Trace and Hilbert 90.
     Apr 18: Cyclic extensions.
     Student presentations on April 18 from 3:30 to 5:30 p.m.
16. (Week of Apr 21 = Dead Week.)
     Apr 21: Cyclotomic extensions.
     Apr 23: Cyclotomoic extensions.
     Apr 25: Review.
     There will be no homework for this week.
17. (Week of Apr 28)
     May 1: Office hour 2 p.m. to 5 p.m.
     May 2: Final exam 8 a.m. to 9:50 a.m.
Some handouts: