Messages
Ark6 on the web. G
The final curriculum is:
Chap. I: Sections 1-6
Chap. II: Sections 8-11
Chap. III: Sections 13-16
Chap. IV: Sections 18-23
Chap V: Sections 26, 27
Chap VI: Section 29
Chap VII: Sections 36, 37
Chap IX: Sections 45 until Cor 45.18 page 395
46 until Cor 46.5 page 402
Geir
In problem 1c) the constant a is supposed to be 3.
In problem 2b): Suppose that p+1 is not congruent 0 mod 5
In problem 2c): Suppose that p and q are primes such that p+1 is not congruent 0 mod q.
Geie
Today I did what was left of section 27, i.e. the paragraphs called "Prime fields" and "Ideal structure of F[X]". Then we did most of section 28, but not theorem 29.18.
Next wednesday, I ll do the first part of section 31, i.e. the paragraph called "Finite Extensions" – The paragraphs "Algebraically closed Fields and Algebraic Closures" and "Proof of Existens of an Algebraic closure" will not part of curriculum , we also must take section 31 out the curriculum.
Then I 'll do a short version of 45, 46 and 47. Details not yet clear.
Geir
Solutions of the mandatory assignment are posted.
G
There was a misprint in problem 26 ( 5 in stead of 7). New version on the web.
G
Today I finished section 23 – Factorization of Polynomials over Fields.
Next week we start on 26 – Ideals and Factor rings – and I guess wil do section 26 and 27, may be start on 29.
Geir
On wednesday I did what was planed: Sections 18, 19 and 20.
We defined rings and fields and integral domains. Proved some elementary properties. Spoke about zero divisors and nilpotent elements. Gave a lot og examples, and studied the rings Zn of residue classes mod n. We proved Fermat's little theorem and Euler's generalization.
G
There is an error in problem 15 on Ark6. The problem should read: Show that all groups of order 15 or 77 are abelian. How many such groups are there?
Geir
On wednesday I started on the Sylow theory, section 36, and did Sylow's first and second theorem. Next week we continue with the third one and do some applications, that is section 37.
Friday we do the exercises.
Geir
There was a few inaccuracies in the note about the alternating groups being simple. Two errors in the proof of theorem 2. A misprint in problem 5 a) and a stupid hint for problem 5 b). Corrected version on the web.
I also have split problem 5 b) in two parts, to make it easier to give a hint.
Geir
I have posted a list of what I plan do in the continuation of the course.
Geir
To day I did what was left of sections 13 and 14 about Homomorphisms and Factor groups.I also did most of section 15, until 15.14, definition of a simple group.
On friday we'll dod the exercises, and probably have time for some theory. In case, I'll Inner automorphisms (last part of on page 140/141) and The center and commutator subgroup (last part of section 15, page 150/151).
Next wednesday we do section 16 (Group action on a set), and if time allows, we jump to section 36, Sylow theorems.
Geir
Probably next week I'll use some time to give a proof of theorem 15.15 which says that the alternating groups on n letters are simple if n is not 4. This is a most important theorem, in fact implying Abels theorem on the quintic equations, and I think it should be part of the course. I also prefer to do it a little differently than as in exercise 39 that book proposes, so I have written up a small note which you find on the web-side of the course.
Geir
NB– The Mandatory Assignment: The problems for the mandatory assignment are posted on the web. The deadline is Thursday March 14 at 14.30. The assignment is to be handed in in the box on the 7. floor of Niels Henrik Abels hus.
Geir
I had no time to do theory to day, so we continuo with Th. 4.11 etc on wednesday. I'll also do some examples of computing with factor groups, e.g., Ex 15.7, 15.9 og 15.10
Geir
NB There will be no lecture on Friday March 22!
I am in America.
Geir
Yesterday I did what was left of section 11, that is paragraph called the Structure of finitely generated abelian groups. Then section 13 about Homomorphism, and I started on section 14 called Factor Groups. I spoke about normal groups again, and did theorem 14.13. We defined factor groups an did theorem 14.9.
I am not sure of how much time we will spend on the exercises on friday. Several of them have already been treated in the lectures (12,13). I plan to do some theory, that is Theorem 4.11.
Next wednesday we do what is left of section 14 (not much) and section 15 and start on 16.
Geir
NB: The mandatory assignment
The time for the mandatory assignment is approaching. The deadline is Thursday March 14. I'll put the problems on the web 14 days before, that is on Thursday February 28.
Geir