### Class Diary for M351, Spring 2007, Jochen Denzler

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Wed Jan 10: ** Intro: Definition of a group and first examples.
*Hwk: study Examples and Hwk 3-6 from the Group Zoopark sheet for
discussion Fri. *

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Fri Jan 12: ** Left neutral and left inverse imply existence and
uniqueness of neutral and inverse. Subgroups. More examples. *Hwk:
7-10; 8 for grading; due Wed.*

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Mon Jan 15: ** MLK DAY

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Wed Jan 17: ** Discussion of group examples; in particular the history
of groups as transformation groups. *Hwk: 10 for grading due Fri.
All others on your own; presentation on Fri as needed* -- Definition
of rings and fields. *Hwk: compare field axioms, as built up from group
and ring in class, with field axioms for the real numbers in M300*

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Fri Jan 19: ** Hwk discussion. First few examples of a ring.
*Hwk: begin 11; more in Monday's class. Read book Sec 4.3*

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Mon Jan 22: ** More Hwk discussion. Examples of Rings. *Hwk 11 due Wed
(for discussion and presentationa as needed, minus formal grading).
Begin to look at new hwk as well*

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Wed Jan 24: ** Discussion Hwk 11. Divisibility, units and notion of
gcd defined in commutative ring with identity; primary example is ring of
integers. Other examples will follow later. *Hwk: 13-15 due Mon (hard
deadline) for grading. 12,16,17 due Wed for grading (soft deadline)*

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Fri Jan 26: ** Euclidean algorithm in the ring of integers.

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Mon Jan 29: ** Prime numbers in the ring of integers, and general remarks
towards generalization of primality / irreducibility to more general rings.
Equivalence of primality and irreducibility in Z. *Hwk: 19 due Wed (not
for grading)*

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Wed Jan 31: ** Hwk discussion. Congruence mod n.
Outline of construction of the ring Zn

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Fri Feb 02: ** Comments on construction of ring R[i] (hwk 15). Proofs for
well-def'dness and ring properties of Zn. *Hwk due Wed: 18,21 not graded
(presentation as needed); 23+24 graded. 20 pingpong correct by Wed= 1pt EC.
Also have a look at 22 and see if you have questions*

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Mon Feb 05: ** Direct sum of rings; Chinese Remainder Thm; zero divisors
and integral domain defined.

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Wed Feb 07: ** finished Chinese remainder theorem. Uniqueness of
prime factor decomposition. Zn is a field if n is prime, and has zero
divisors if n is composite. *Hwk: 22, 25-27 for presentation as
needed. (Note: formerly numbered funnily: 22,25,25,26)*
Begun: The chinese remainder theorem translated into
an isomorphism of rings statement. (with the isomorphism notion yet to
be defined during, and motivated by this example).

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Fri Feb 09: ** Ring (and group) homomorphisms and isomorphisms; in
particular the example Zm + Zn isomorphic Zmn for gcd(m,n)=1. Euler-phi
function introduced.

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Mon Feb 12: ** Discussion of Hwk 25. Another example of a ring
isomorphism (from Example + Hwk 36)

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Wed Feb 14: ** Review of Hwk 16. Isomorphism of P(M) with a ring
of functions via indicator function *Hwk: 31-41 for presentation as
needed*

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Fri Feb 16: ** More homo-/isomorphism examples. *Hwk 40 for grading
due Wed*

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Mon Feb 19: ** Euler-phi; multiplicativity property and formula.
a to the power phi(n) congruent 1 modulo n, provided gcd(a,n)=1.
period lengths of fractions in decimal expansion.

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Wed Feb 21: ** Proofs concerning period lengths of 1/p and phi(p).
Other uses of modular arithmetic. *Hwk: 28-30 for grading due Mon;
44 already by Fri (no grading)*

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Fri Feb 23: ** comments on Hwk 44. Ideals; defs and examples.

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Mon Feb 26 :** Hints for Hwk 29, 46. --- Hwk 28-30 and 46 will
be collected for grading
on Wed; hard deadline; and solutions posted then, for exam prep use.
Ideals are for the purpose of constructing residue class rings.

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Wed Feb 28: ** Construction of residue class ring with proofs. Example
Z[i]/(3) started.

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Fri Mar 02: ** EXAM 1

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Mon Mar 05: ** Z[i]/(3) finished. Exam back. *Hwk 42,45 for grade asap.
43 for discussion as needed*

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Wed Mar 07: ** Overview over polynomial rings: Modular arithmetic helps
to decide factorization of polynomials with rational coefficients;
therefore we study also polynomials whose coefficients are in rings Z_n.
But in those rings, polynomials must be distinguished from polynomial
functions. So we require a pure-bred algebra construction of polynomial
rings, not relying on polynomial *functions* as in calculus.

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Fri Mar 09: ** Construction of R[[X]] (ring of formal power series)

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Mon Mar 12: ** SPRING BREAK

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Wed Mar 14: ** SPRING BREAK

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Fri Mar 16: ** SPRING BREAK

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Mon Mar 19: ** associativity of . in R[[X]] proved. The polynomial
ring R[X] as subring of R[[X]]. Degree defined. *Hwk: 50-53 due Fri
for grading (hard deadline)*

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Wed Mar 21: ** Properties of the degree. The euclidean algorithm for
polynomials (started).

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Fri Mar 23: ** Existence of a gcd in R[X] for R a field; begun. R[X] is a
principal ideal domain.

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Mon Mar 26: ** Divisibility, units in integral domains; examples; a gcd
exists in each PID. *Hwk: 47,48 due Fri for grading*

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Wed Mar 28: ** Comparison of `pedestrian' and PID proof of existence
of gcd. Unique factorization begun.

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Fri Mar 30: ** Proof prime factorization

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Mon Apr 02: ** Uniqueness of prime factorization proof finished. *Hwk:
54-57 due Mon (no grading; presentation as needed). 58-60 due Wed after
Easter for grading. Sol's will be expected to be careful, 300 style. *

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Wed Apr 04: ** Quick overview of book's thm:``If the Integral domain R is
a UFD, then R[X] is a UFD, too'' -- We'll do this for Z[X] whose UFD
property is still in question b/c it is not a PID. The method relies on
knowing that Z is a subring of the field Q, and Q is the samllest field
containing Z as a subring. Such a field can be constructed for every
integral domain (the book did it, we didn't, yet). So we'll prove the UFD
property for Z[X], but the proof is paradigmatic for the book's general
case.

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Fri Apr 06: ** GOOD FRIDAY

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Mon Apr 09: ** Gauss lemma; factorization in Z[X]

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Wed Apr 11: ** Uniqueness of factorization in
Z[X]. -- Evaluation homomorphism; roots; polynomials vs polynomial
functions.

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Fri Apr 13: ** Irreducibility tests: rational root test; Eisenstein

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Mon Apr 16: ** EXAM 2

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Wed Apr 18: ** Discussion of exam. Factor rings of polynomial rings
(review and examples).

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Fri Apr 20: ** More examples for factor rings of polynomial rings.
Evaluation

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Mon Apr 23: ** Z_2[X]/(X^2+X+1) is a field. Informally: General
construction principle for fields with finitely many (prime power)
elements is: Mod out an ideal generated by an irreducible polynomial
of degree n from Z_p[X]. (No proofs given). Review on some properties of
polynomial rings with this hwk in view: *Hwk: 61-65; won't be graded;
sol's to be posted soon*

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Wed Apr 25: ** R comm ring with 1.
When is a factor ring R/I a field? Answer: exactly when I is a maximal
ideal. Maximal ideals in poly rings F[X] are exactly those generated by
irreducible polynomials.

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Fri Apr 27: ** Q&A; in particular on Euler-phi. Some comments on exam.
A quick overview how to construct the quotient field of an integral domain.

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Mon Apr 30: ** STUDY PERIOD. Regular class time will offer a Q&A session.

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Wed May 02: ** FINAL EXAM: 10:15-12:15
*
scheduled by university policy *

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