The time now is Sun 05 Jul 2015, 13:42
All times are UTC  4 
Author 
Message 
mahaju
Joined: 11 Oct 2010 Posts: 493 Location: between the keyboard and the chair

Posted: Sun 20 Mar 2011, 08:16 Post subject:
What does this maths symbol mean? 

This is an excerpt from an article I am reading. In the 2nd line the Z refers to set of integers I think, but what does Z/m mean? Does it mean the set {0,1, ..., m1} (ie, the set of integers mod m) or does it mean something else?
Please help me out here.
Thanks in advance.
Also, a homomorphish which is one to one and onto (bijective) is called isomorphism. Is this true?
Description 

Filesize 
33.48 KB 
Viewed 
778 Time(s) 


Back to top



myke
Joined: 15 Mar 2011 Posts: 102 Location: Québec

Posted: Sun 20 Mar 2011, 10:45 Post subject:
The Answer Subject description: What Z/m means 

In mathematics, Z/m indeed refers to the set {0,1, 2, ..., m1} but generally mathematicians are more interested in looking at abstract groups, rings and fields. The group {Z/m,+} refers to the set Z/m with addition modulo m. So, the subgroup {Z/4,+} would be {0,1,2,3} with 3+1=0, 1+2=3, 2+3=1, etc. You can think of it as adding the remainders of two numbers so the remainder of 11 after dividing by 4 + the remainder of 9 after dividing by 4 = the remainder of 20 after dividing by 4. It forms a commutative group with the identity being 0. The meaning I gave it is in some sense canonical. (The technical term is isomorphism).
If you look at {Z/m,+, ⊗} , with the addition and multiplication being modulo m, you get a commutative field with the additive identity being 0 and the multiplicative identity being 1. In other European languages, field is generally not used in this context, French: corps, German: Körper, Dutch: lichaam, but Flemish: veld.
_________________ AA1 D255Ekeucr slacko 5.3;luci;mijnpup; twos; with:Emacs,gawk,noteboxmismanager,treesheets, freeplane, libreoffice, tkoutline, Sigil, calibre, calendar. magic&Noteliner(wine), kamas (DOS)

Back to top



mahaju
Joined: 11 Oct 2010 Posts: 493 Location: between the keyboard and the chair

Posted: Sun 20 Mar 2011, 21:02 Post subject:


Thank you very much
Considering the symbol to mean the set Z mod m the other parts of the article make a little more sense, but I still have many problems with this. For instance, do you have any idea what it means in line 5, that
Steps 1 and 3 are "noops" if Z/m is represented as {0,1,...,m1}
What is "noops" and why the "if"? ( since Z/m means {0,1,...,m1} )
Again, in this next section, R[x] means the set of all polynomials in R with variable x, that is, the coefficients of the variables in the polynomials are members of the ring R. So does that mean R[y][x] (line 2) is the set of polynomials of two variables x and y? And what is the difference between writing R[y][x] and R[x][y], because in later sections the articles hunts that they are not the same, however it's not clear how they are different. Also, by analogy, R[y][x]/(x^ny) would mean a set of polynomials that we get as remainder when we divide the members of the set in the numerator by the denominator. Is this right? Do I have to visualize this new set as something being finite like Z/m, or is it possible for this new set to have infinite members?
Thanks

Description 

Filesize 
35.94 KB 
Viewed 
294 Time(s) 


Back to top





You cannot post new topics in this forum You cannot reply to topics in this forum You cannot edit your posts in this forum You cannot delete your posts in this forum You cannot vote in polls in this forum You cannot attach files in this forum You can download files in this forum

Powered by phpBB © 2001, 2005 phpBB Group
