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Curriculum for MAT4250:
Officially the curriculum is from the book (Algebraic Number Fields by Gerald J- Janusz). To make it simple: The whole first chapter, and the second chapter until the top of page 106, that is, Corollary 3.7 is included.
Practically, my notes are the curriculum.
Good luck! Geir
And again: Do not hesitate to contact me if there are questions or issues.
1) A new and fairly compete version of "The class number formula" is posted. All I said yesterday was not correct. Check with the notes.
2) No Lecture on Tuesday 26.
3) Three hours of lectures on Thursday 28. From 14.15 to 17.00!
And that will be the last lecture.
Geir
1) A new and fairly compete version of "The class number formula" is posted. All I said yesterday was not correct. Check with the notes.
2) No Lecture on Tuesday 26.
3) Three hours of lectures on Thursday 28. From 14.15 to 17.00!
And that will be the last lecture.
Geir
I have put a new and much better version on the web. It is still incomplete in that the volume computation is missing. Hopefully I ll be able to write t down soon.
G
Since you asked for notes and since I don not follow the book,
I have posted what I have written so far about Zeta. But be aware, it just a raw sketch (with all kinds of errors) and by no means complete. Hopefully, it is useful though.
Geir
The lecture on next thursday, that is Nov 7, is moved to wednesday Nov 13, at 10.15. Room will come.
Geir
I have posted a very preliminary version about Dirichlet's theorem. For some reason I stared to write in Norwegian and did not realise it until I was more half way three. Sorry for this if some non norwegian speaking students still are interested. The proof is, of course, up to presentation, the same as in the book.
It is a very raw version, but I hope it will be useful for you though.
Geir
I posted notes about Minkowski bounds etc. It covers about what we did last week and what we will do today. Warning: They are very preliminary, but hopefully they will be useful for you.
Geir
I have posted a new and extended version of the notes about cyclotomic fields including a proof of the law of quadratic reciprocity.
G
Have posted a preliminary version of notes about cyclotomic fields
G
I have uploaded some notes about norms, which we are going to do to morrow.
To morrow we finish the example on page 41.
We do paragraph 8 (norms)
We skip the paragraph 9, the last part about quadratic extensions we have done long time ago.
Then if time allows, we start on paragraph 10 – cyclotomic extensions.
Next week we finish paragraph 10, we do paragraph 11 about quadratic reciprocity. And I guess that will be what we have time to do.
Geir
I have posted a another new version og Extensions on the web, where the errors we discovered during the lecture are corrected.
There is no lecture on Tuesday October 1.
On Thursday September 26 and Thurday October 3. I'll be a way, but Kristian Ranestad will be a substitute.
When I am back, on Tuesday 8. I'll do problem 11 in the notes on Extensions on the board.
Geir
I have put a new version of the notes about extensions on the web.
I have included some more exercises. For the two last exercises we have not yet all theory needed, but the rest should be within what we have done.
Geir
Yesterday I started on section 6, Extensions of Dedekind rings.
Tomorrow I'll finish the proof of the main theorem, basically corollary 6.7 in the book. If time allows we also do Galois case and may be start on section 7, Ramification and the Discriminant.
Geir
Have posted a first and raw version of the notes on GT.
G
The lecture tomorrow is moved to Friday at 14.15 to 16.00. It takes place in seminar room B91.
This applies only to the lecture this week.
Geir
On thursday I finished the chapter about Dedekind rings. To morrow I'll speak about norms and traces section I.5.
NB.: The lecture this thursday (that is sept. 5) is moved to wednesday 10.15-12 in B91.
I'll speak about separable extensions, an do some Galois theory.
Geir
I remind you that we have moved the lectures on Wednesdays to Thursdays from 14.15 to 16.00.
New location: Seminar room B81.
Geir
I have put a new version of the notes about Dedekind rings on the web. Some errors are corrected and it is somehow expanded.
G
To day I finished the section about integral dependence.
I did the basics about fractional ideals.
I gave an example of a non-invertible ideal, and calculated the integral elements in the quadratic number fields.
Next week we start on DVRs and Dedekind rings.
I have posted two small notes, one in Norwegian with some basic algebra, and one in English about Dedekind rings.
Geir
Today I did parts of section 2 in the book, Integral dependence.
I 'll do example 1, 2 and 3 on Thursday. We postpone 2.5.
I ll prove that being integrally closed is a local property.
Then I prefer to speak about fractional ideals (i.e. first part of section 4) before we start on section 3 with DVRs and Dedekind rings.
We have permanently moved the lecture on wednesday to thursday at 14.15 to 16.00. Room will come.
Geir
The lecture on Wednesday, August 21 is moved to Thursday, August 22. New location: Seminar room B63 ( NHA 637 ).
Geir
Welcome to MAT4250!
The text we shall be using is the book: "Algebraic Number Fields" by Gerald J. Janusz; Graduate Studies in Mathematics, Volume 7.
In principle I will follow the text rather closely, but surely in my own way. I am also open for suggestions from you, and of course, if there is some background stuff that most of you need to refresh, we will see to that in one way or another.
We certainly will need some Galois-theory which is not covered in the book, and I guess we will use some time on refreshing that though ot at the beginning of the course. I nice text about Galois theory is J.S. Milne's "Fields and Galois theory":
http://www.jmilne.org/math/CourseNotes/ft.html
Geir