### Class Diary for M351, Fall 2004, Jochen Denzler

**
Mon Jan 12: ** Intro; repetition of field axioms from 300;

--->
Hwk1.pdf. *Do Problem 2 a-c for hand-in by Wed.
Part 3 of Problem 1 for discussion on Wed. *

**
Wed Jan 14: ** Why axioms are called their respective names.
Def'n of a ring etc. Examples. *Do problem 2 d,e for discussion on Fri;
try 2f already, preparing for later assignment of it. Study pp 6-9. *
Info: fields are discussed on pp 75-80 (or -84),
but do not expect to understand pp 69-74 yet.

**
Fri Jan 16: ** Discussion of hwk. Distinction of 0 as the `individual'
number zero, as compared with the general object called 0 in the axioms, and
other foundational issues of what our words and symbols mean.
* Think about 2(f) and (g) over the weekend; no formal turn-in scheduled
yet. *

**
Mon Jan 19: ** MLK DAY

**
Wed Jan 21: ** Simple thms for rings (basically pp 14-19); efficient
checking of all ring properties (using table inspection where feasible) in
Ex. 3.4 on p.7; *Hwk 2fg from handout due as
pingpong hwk (2pts)
by Friday. Also from p.10ff: #1e-g and #12
by Friday, and #17 by Monday (5 x 2pts)*

**
Fri Jan 23: ** efficient checking etc finished. Ring properties of
the power set with symmetric difference and union.
Hwk2.pdf. *Do problems 3,4 by Wed, 5 by Friday*

**
Mon Jan 26: **More hints on pingpong hwk;
Begun discussion ``new rings from old''
*Do problems 6,7 also by Friday; Problem 8 by Monday*

**
Wed Jan 28: **Discussion of questions about hwk
*Assignment of #3,4: time extended til Friday*
Handed out an updated version of Hwk2.pdf:
typo in the printed version (fixed online);
no assignment yet

**
Fri Jan 30: ** Discussion of questions about hwk;
Basic examples for ``New Rings from Old'';
*Problems 9,10,11 due Wednesday*

**
Mon Feb 02: ** Ordered Rings (def'n and immediate consequences);
repetition of induction principle.

**
Wed Feb 04: ** multiples and powers defined in a ring; formal proof
of (m+n)a = ma + na given *Problems 12,13,14,20 due Monday*

**
Fri Feb 06: ** Questions; finished up induction properties; defined
congruence mod n * Problems 16,18 due Wednesday; 17, 19 due Friday*

**
Mon Feb 09: ** Congruence mod n in the ring of integers;
congruence classes; division with remainder
**Solutions to Pblms 3-20 posted here
(readable thru 14; rest to become readable in due time by Friday)**

**
Wed Feb 11: ** Division with remainder finished up;
Ring **Z**_{n} defined; issue of well-definedness explained
and ring properties proved.
*Problems 21,22 due Monday; 23,24 due Wednesday*

---> Hwk3.pdf

**
Fri Feb 13: ** Discussion of hwk; divisibility by 9; def'n of primes;
(Note: now jumping from p48 to p93 in the book)

**
Mon Feb 16: ** GCD and Euclidean algorithm. Why we want ``s|d'' in the
definition, not ``sHwk deadline for 21,22 extended till Wednesday. Also, you may hand in
22 or 24 as pingpong homework, if you feel you need more help than the
hints and discussion from class offered

**
Wed Feb 18: ** Euclidean algorithm proved, and ``p|ab => p|a or p|b''.
What's on the exam? -- *Problems 25-27 due Monday (File Hwk3 has been
updated with more pblms)*

**
Fri Feb 20: ** Discussion of pblm 24 mainly. More hints for this pblm
here

**
Mon Feb 23: ** Applications of modular arithmetic
**Solutions to Pblms 21-27 posted here
**

**
Wed Feb 25: ** FIRST IN-CLASS EXAM

**
Fri Feb 27: ** The unique factorization into primes, and the relevance
of the euclidean algorithm in its proof; Non-unique factorization in the
ring of even numbers.
* Hwk for Mon: get finished with 24; new 31. For Wed: 32.
For Fri: 28,29,30 (File Hwk3.pdf has been
updated with more pblms)*

---> Handout fermat.pdf
tells about how/why rings like **Z**[i] were born

**
Mon Mar 01: ** Exam back. **Z**_{n} is a field if n is prime,
and has zero divisors if n is composite. The Euler phi function

**
Wed Mar 03: ** Units in a ring; Eulers' thm and little Fermat;

**
Fri Mar 05: ** Homomorphisms and isomorphisms (that's p. 48-53 in the
book);
in particular the isomorphism from **Z**_{mn} to
**Z**_{m} (direct sum) **Z**_{n} . Consequence for
Euler-phi.

**
Mon Mar 08: ** SPRING BREAK

**
Wed Mar 10: ** SPRING BREAK

**
Fri Mar 12: ** SPRING BREAK

**
Mon Mar 15: ** Review on powers in **Z**_{n}
and modular arithmetic.
*Hwk 33,34 by Friday, 35-38 by Monday next week
(File Hwk3.pdf has been updated with more pblms)*

**
Wed Mar 17: **Practical algorithm for computing powers modulo (large)
numbers (see here for typeset sample);
a few examples on homo-/isomorphisms

**
Fri Mar 19: ** Discussion of upcoming hwk. More homomorphisms and isomorphisms.
*Hwk 39-43 due later next week; but start them now so we
can discuss questions that may arise;
see file Hwk4.pdf*

---> **Solutions to Pblms 21-38 posted here
(latest links to become active by Monday afternoon)
**

**
Mon Mar 22: **Abundance of homomorphisms and isomorphisms:
complex conjugates; its analog for Z[sqrt(2)] -> Z[sqrt(2)] ---
evaluation of functions at a point; composition (change of independent
variable); viewing numbers as constant functions ---
masking 2x2 matrices as 3x3 matrices by filling in zeros.
*Hwk 39-43 due
Friday; if questions arise, ask Wednesday already*

**
Wed Mar 24: ** final examples of homo- and isomorphisms; THEC survey;
questions on hwk.

**
Fri Mar 26: ** Polynomial rings and rings of formal power series

**
Mon Mar 29: ** Polynomials vs polynomial functions; the evaluation
homomorphisms; and the possible non-injectivity of the homomorphism from
polynomials to polynomial functions. Divisibility and Irreducibility
for polynomials

**
Wed Mar 31: ** The euclidean algorithm, and the gcd in F[X]
*Hwk: 44 by Friday; try to get 45 done by Friday; if needed
I'll extend it till Monday. (Hwk4.pdf updated)*

---> Solutions to Hwk 39-43 posted here

**
Fri Apr 02: ** Quick discussion of exam subjects; example for gcd in F[X].
*Hwk: 45 by Monday (if not handed in yet). Hwk 46,47 won't be graded,
but are exam relevant, so do them by Monday. I'll post solutions and answer
questions then. ***NOTE:** I have changed the polynomials in Hwk 46,47
to get less tedious numbers in 46. But either version is good for training.

**
Mon Apr 05: ** Monic polynomials; roots of a polynomial: examples & thm

**
Wed Apr 07: ** EXAM NUMBER 2

**
Fri Apr 09: ** GOOD FRIDAY

**
Mon Apr 12: ** Irreducible vs prime: distinct in a general integral
domain, but equivalent in the presence of a euclidean algorithm.
Roots and irreducibility (proof of thm).

**
Wed Apr 14: ** exam back; *Hwk 48,49 and exam pblm 8 for Friday;
50-52 for Monday*

**
Fri Apr 16: ** Discussion of exam pblm 8. A bit on ideals.

**
Mon Apr 19: ** Ideals and Factor Rings (generalizing our construction of
**Z**_{n}): the commutative case only

---> More hwk solutions posted here.
Exam posted here

**
Wed Apr 21: ** Factor rings of polynomial rings; construction of
**C** out of **R** by modding out from **R**[X]
the ideal consisting
of all multiples of X^{2}+1 from **R**[X].
*Hwk 53-56 is on updated version of Hwk4.pdf;
we won't grade it, but I'll post solutions*

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Fri Apr 23: **

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Mon Apr 26: **

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Wed Apr 28: **

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Fri Apr 30: ** STUDY PERIOD

**
Tue May 04: ** 10:15-12:15 FINAL EXAM