Re: Significance of Prime No in Cyclotomic Integers

From: James Buddenhagen (foo_bar_at_texas.net)
Date: 03/29/05 

Date: Tue, 29 Mar 2005 15:02:35 GMT

"Sundar Krishnan" <sundark100@yahoo.com> wrote in message
news:11108311.1112021993521.JavaMail.jakarta@nitrogen.mathforum.org...

[Sundar's message deleted]

Based on your questions a broad brush not very accurate overview might be
helpful to you. I will not answer your questions, just try to give the
overview.

Roughly: a system of 'numbers' in which you can add, subtract, multiply and
divide and get answers within the system is called a 'field'. If you can do all
of the above but division sometimes gives answers outside the system then
you are looking at a 'ring'.

The rational numbers are a field, the ordinary integers are a ring.

Other fields: the real numbers, the complex numbers, all numbers of the
form a+b*sqrt(2) where a,b are rational, all numbers of the form
a+b*sqrt(-3) where a,b are rational is a field.

Note that the last two examples of fields are associated with the irreducible
polynomials x^2 - 2, and x^2 + 3. (irreducible means they don't factor
with factors having integer coefficients). All the above examples of fields
are subfields of the complex numbers.

This suggests (perhaps) that whenever we have an irreducible polynomial
with integer coefficients there might be a subfield of the complex numbers
associated with it, which contains all roots of the polynomial. This is the
case. Such fields are called "algebraic number fields". Their study is
called "algebraic number theory". Inside each algebraic number field
sits a ring of numbers called 'algebraic integers' that behave in some
ways like ordinary integers. One can factor some of them. But is such
factorization unique?

If you restrict your attention to fields that come from the irreducible
factors of x^2-1, x^3-1, x^4-1, x^5-1, etc, then you are looking at the
cyclotomic fields. [So called because the roots (complex numbers)
of such polynomials cycle around the unit circle]. BTW, roots of x^n -1
are called 'roots of unity'. I have never heard a human call them 'De Moivre
numbers'.]

So, if we call the numbers in a cyclotomic field, 'cyclotomic numbers' then
the ring of integers in that field are 'cyclotomic integers'. In that ring
there
are special numbers called primes, and one can factor cyclotomic integers
into products of primes and ask is that factorization unique. And the answer
is (in contrast to ordinary integers): not always.

May I suggest you find a library with math books and read. The wolfram
site is not the best place to learn this stuff. You might find the following
old book on this stuff interesting. It is available on line:
http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=01200001&seq=3

Be warned that terminology (and notation) has changed. The book's
"number realm" is now called "number field". Good luck.

Jim Buddenhagen

Relevant Pages

  • Re: JSH: Same old crap
    ... f is not coprime to them in that ring. ... The fact is that every number coprime to any of the roots will be coprime to all the roots. ... any such ring must fail to include all roots of integral polynomials for which it contains one root. ... properties than the roots of irreducible polynomials. ...
    (sci.math)
  • Re: Some math, algebraic integers
    ... >>Note that there are plenty of reasons for the roots of irreducible ... >>be related to the coefficients of their irreducible polynomials. ... he supposed it was true in the ring of integers. ... when going beyond the ring of algebraic integers. ...
    (sci.math)
  • Re: Significance of Prime No in Cyclotomic Integers
    ... I am aware of Cube Roots of Unity. ... > you are looking at a 'ring'. ... > of such polynomials cycle around the unit circle]. ... > is (in contrast to ordinary integers): ...
    (sci.math)
  • Re: JSH: Inconsistency with algebraic integers
    ... in the ring of algebraic integers. ... And which of the roots ... *Reducible* polynomials have to be factored ...
    (sci.math)
  • Re: JSH: Now a change
    ... > intersection of two rings forms a ring. ... > necessity, some algebraic numbers for which the minimal polynomial ... can add one of the roots to the ring and you get the factorisation ... The problem may crop up when you try to add some roots of other polynomials ...
    (sci.math)