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2012-05-14, 10:14   #1
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

722110 Posts
Tryin to edify myself...

http://modular.math.washington.edu/e...ml/node17.html

(I recently figured out that the author of that page is the same Stein who wants B200 factored. )

In the meantime:

In the link, he's trying to introduce your average ring modulo n. He starts by defining groups and rings, a very sensible thing to do.

(There are missing images, but if you really want to see them, just hit the link.)

Quote:
 In this section we define the ring of integers modulo , introduce the Euler -function, and relate it to the multiplicative order of certain elements of .
Alright, good, I assume he'll define "multiplicative order" later.
Quote:
 If and , we say that is congruent to modulo if , and write .
Sure, basic modular arithmetic.
Quote:
 Let nZ = (n) be the ideal of Z generated by n.
Wat?!
He defines what a ring/group are, defines what modulo congruence is, but expects me to know what an ideal is?
Quote:
 Definition 2.1 (Integers Modulo ) The ring of integers modulo is the quotient ring of equivalence classes of integers modulo . It is equipped with its natural ring structure:
I understand what a ring is, I can guess what a quotient ring is (should define that too) and I have no clue what "equivalence classes" mean. (Again, I can guess, but having more definitions in there would be freaking awesome. Later note: He sort of implicitly defines it on the next line, but still.)
Quote:
 We use the notation Z/nZ because Z/nZ is the quotient of the ring Z by the ideal nZ of multiples of n.
Again, why does he think I know anything about rings or their quotients if he only just defined it for me?
Quote:
 Because Z/nZ is the quotient of a ring by an ideal, the ring structure on Z induces a ring structure on Z/nZ .
Ditto for "ring structure". If he's defining a ring, why does he not define this? (He does have some picture-tex-equations, but they don't serve as a definition and it's not clear precisely what they mean; I can only make educated guesses (as it were).)
Quote:
 We often let a or a mod n denote the equivalence class a+nZ of a.
And there's the only help I see about equivalence classes.

And then he goes on to define a field. Ironically, groups and fields are the only things I really knew before hand, and those (and rings, which are basically the same concept) are the only things he bothers to define.

Now, before Dr. Silverman jumps on me, I know there are plenty of other places to learn these things, but I have yet to find one to my liking. Any recommendations? I tried to Wikipedia ideals last time I asked a stupid question (see yafu bugs thread concerning NFS for Alq4788), but it really made no sense at the time, so I'll give it another shot tomorrow (as with the other as-yet-undefined terms). Any insight here is appreciated, though I'm more looking for references to other resources (besides Wikipedia) which contain (hopefully educational) definitions rather than you guys just answering (but I'll take whatever anyone's willing to give ).

Alternately, if someone can help be get a grasp on what a ring structure (or the other things) are, as in help me understand them as opposed to just staring at a definition and saying "Okay, sure", that would be appreciated as well. This is at the moment a somewhat secondary goal though.

(I posted this because it's somewhat frustrating; looking at Stein's table of contents, especially what the first section is all about, it gives the impression that the work was more-or-less self contained. It's now apparent that it isn't -- though I'm still willing to believe that this particular page was an accident and that the rest is self contained.)

Last fiddled with by Dubslow on 2012-05-14 at 10:18 Reason: s/typical average/average/

2012-05-14, 11:28   #2
Gammatester

Mar 2009

1001102 Posts

Quote:
 Originally Posted by Dubslow http://modular.math.washington.edu/e...ml/node17.html ... Alternately, if someone can help be get a grasp on what a ring structure (or the other things) are, ...

There is a free pdf version of Stein's "Elementary Number Theory" at http://sage.math.washington.edu/ent/ and it lists the backgound:
Quote:
 Background. The reader should know how to read and write mathematical proofs and must have know the basics of groups, rings, and fields. Thus, the prerequisites for this book are more than the prerequisites for most elementary number theory books, while still being aimed at undergraduates.
A good text with all (or most) terms properly defined is Victor Shoup's "A Computational Introduction to Number Theory and Algebra":
Quote:
 A book introducing basic concepts from computational number theory and algebra, including all the necessary mathematical background:
A pdf of version 2 is available from http://shoup.net/ntb/

Last fiddled with by Gammatester on 2012-05-14 at 11:40

2012-05-14, 11:54   #3
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

20C016 Posts

Quote:
my source of information is even older I believe (from the number theory thread I started) but defines equivalence class ( based on equivalence relations):

Quote:
 If ∼ is an equivalence relation on S, then for x ∈ S one deﬁnes the set [x] := {y ∈ S : x ∼ y}. Such a set [x] is an equivalence class
I've of course bolded equivalence relation because it's the next lowest thing to learn about.

Last fiddled with by science_man_88 on 2012-05-14 at 11:56

2012-05-14, 19:53   #4
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3·29·83 Posts

Quote:
 Originally Posted by Gammatester You can start with a newer version, the page you reference is almost five years old! There is a free pdf version of Stein's "Elementary Number Theory" at http://sage.math.washington.edu/ent/ and it lists the backgound: A good text with all (or most) terms properly defined is Victor Shoup's "A Computational Introduction to Number Theory and Algebra": A pdf of version 2 is available from http://shoup.net/ntb/
Okay, thanks. I had actually already found the second PDF, but I remember not liking it for some reason or another... I also prefer HTML to PDFs, 'cause then the author can put buttons in and organize the book much better rather than just being a whole bunch of separate PDF pages back to back. That's one reason I was using the old version of Stein; the other reason is that the Wikipedia page on P-1 links to the old version's explanation of P-1, which is how I found it in the first place.

Edit: Double thanks for the newer edition, it seems at least he defines "ideal":
Quote:
 Let nZ = (n) be the subset of Z consisting of all multiples of n (this is called the "ideal of Z generated by n").
It seems to me that this definition of "ideal" means the same thing as "equivalence class" (in this context, at least).

Last fiddled with by Dubslow on 2012-05-14 at 20:13