
MA522
Computer Algebra Fall 2021 Online on Zoom live or in Dabney 331, Tue&Thur 11:45am1:00pm 
Syllabus  People  Maple  Projects  Homeworks  Reading  Grading  Academics 
Current Announcements

Peoples' home pages: Erich Kaltofen, Maple programs for the course.


Course Outline*  
Lecture  Topic(s)  Notes  Book(s)  

1. Aug 17 
Administrative meeting.
Algorithm Defined.
Undecidability of the Halting Problem.

Robert McNaughton,
Elementary Computability, Formal Languages, and Automata,
Section 1.1.


2. Aug 19 
First algorithms:
Karatsuba's Algorithm;
bigOh notation


GG §4.3, GG §20


3. Aug 24 
Algebraic Random Access Machine (RAM) model of computation;
analysis of Karatsuba's Algorithm;
bit complexity

Ka88_jacm.pdf, Section 2.

GG §4.3, GG §20


4. Aug 26 
Repeated squaring, RSA

3.mws,

GG §2; §3; §5.4


5. Aug 31 
Extended Euclidean algorithm;
Hermite elimination

hermite.mws,
lagrange.mws

GG §3.3; §4.5; §5.2


6. Sep 2 
Analysis of Euclid; Chinese remaindering

chrem.mws



Mon, Sep 6  Labor Day, no class  
7. Sep 7 
Newton and Lagrange interpolation;
ReedSolomon code and decoding

ISSAC 21 talk,
welch_berlekamp.mws

GG §5.10, §5.11


8. Sep 9 
use of interpolation/CRA;
rational number recovery;
continued fraction approximations of a rational number

[Kaltofen and Rolletschek 1989,
Theorem 5.1], KR_ratrec.mpl, KR_ratrec.mws,
cont_frac.pdf,
cont_frac.mws



9. Sep 14 
ReedSolomon decoding by rational function recovery

BW_rat_fun.mws,
notes.pdf

GG §5.8


10. Sep 16 
Pollard rho; birthday paradox

Pollard rho code: new_pollard_rho.mpl,
new_pollard_rho.mws,
new_pollard_rho.pdf

GG §19.4


11. Sep 21 
Primitive elements modulo p;
computing discrete logs via Shanks's babysteps/giant steps method and Pollard rho

Discrete log Pollard rho code:
new_discrete_log.mpl

Teske's paper


12. Sep 23 
Diffie/Hellman/Merkle key exchange; el Gamal crypto system


GG §20.3, 20.4


13. Sep 28 
Distribution of primes

rand_mat_det.pdf,
rand_mat_det.mw

[Kaltofen and Villard
2004, Lemma 4.3, p. 112]


14. Sep 30 
Randomized PIT (polynomial identity testing):
Freivald's matrix product test, Jack Edmond's
singularity test of a matrix with linear forms
as entries; DeMilloLipton/Schwartz/Zippel Lemma


GG §6.9


MonTue, Oct 45  Fall Break, no class  
15. Oct 7 
Definition of intergral domain,
field of quotients;
prime and irreducible elements;
GCDs and LCMs;
UFDs


GG §25.2, §25.3


16. Oct 12 
Euclidean algorithm for polynomials over a field;
Sylvester resultants

sylvester.mws, sylvester.txt.

§6.3


Wed, Oct 13, 23h59  Last day to drop the course 

17. Oct 14 
Sylvester Resultants in roots;
The fundamental theorem on symmetric functions.

sylvester.mw,
sylvester.pdf.



18. Oct 19 
Fractionfree Gaussian elimination;
Fundamental Theorem of Subresultants

sylvester.mw,
sylvester.pdf.

GG §6.10 and §11.2


19. Oct 21 
Algebraic extension fields; construction of a splitting field.

small_tower_of_fields.mw, small_tower_of_fields.pdf


20. Oct 26 
Isomorphism of splitting fields; Galois group; separable and inseparable extensions

small_tower_of_fields.mw, small_tower_of_fields.pdf



21. Oct 28 
Norms and traces;
the ring of algebraic integers

small_tower_of_fields.mw, small_tower_of_fields.pdf,
tower_of_fields.mw, tower_of_fields.pdf.



22. Nov 2 
Cyclotomic extensions; the infrastructure of finite fields

finite_fields_as_splitting_fields.mw, finite_fields_as_splitting_fields.pdf.



23. Nov 4 
Factoring polynomials over finite fields:
the distinct degree and CantorZassenhaus algorithm


GG §14


Fri, Nov 5  Topic for class presentation must be declared at 17h  
24. Nov 9 
Factoring polynomials over finite fields cont.:
the Berlekamp polynomial factoring algorithm; Camion's large primes method

factmodp.mws

GG §14.8


Wed Nov 10  Approvals of topics for term papers by me are posted  
25. Nov 11 
Polynomial ideals;
Hilbert Nullstellensatz


GG §21.1


26. Nov 16 
Algebraic closure of the rational numbers;
Effective Hilbert Nullstellensatz;
properties of Gröbner Bases




27. Nov 18 
Proof of Hilbert Basis Theorem;
definition of Gröbner Basis via leading term ideals;
Buchberger's Algorithm

groebner.mw,
groebner.pdf

GG §21.5


28. Nov 23 
Critical pair/completion paradigm: GCDfree basis construction

[Kaltofen 85, Section 3]



WednesdayFriday, Nov 2426 🦃  Thanksgiving, no class  
Thur., Dec 2, noon3:30pm  Presentations  
Friday, December 10, 11:59am  Fall grades due 
Online information: All information on courses that I teach (except individual grades) is now accessible via html browsers, which includes this syllabus. Click on my courses' page of my resume. You can also find information on courses that I have taught in the past.
There will be six homework assignments of approximately equal weight and a choice of a Maple programming project or a term paper. At the end of the course, each student will give a 30 minute presentation on material from the book not covered by me. A choice of topics will be provided by me. Class attendance will not be monitored in any way. If you need assistance in any way, please let me know (see also the University's policy).
Grade split up  
Accumulated homework grade  40% 
Maple project or term paper  30% 
Presentation  30% 
Course grade  100% 
If you need assistance in any way, please let me know (see also the University's policy).
©2009, 2012, 2016, 2018, 2021 Erich Kaltofen. Permission to use provided that copyright notice is not removed.