Re: field generated by the set of roots of unity


let me try to give a little better motivation

these polynomials can actually be solved in C
(the complex numbers)
and these solutions behave a lot like polynomial solutions

there is a sense in which these generalised polynomials
have "zeroes" in number
(on a given branch)
corresponding to a "depth" or "level"
of the exponents
(by an application of cauchy's theorem
and some algebraic trickery)

these "zeroes" are kind of weird
when compared to polynomials
because they have a lot more structure

and most of these zeroes are not algebraic

so looking at these extensions
some W_n[x] / (g) extension over W_n
seems an accessible target of study
in that world beyond algebraic extensions of Q

it's a manageable transcendental field
and an interesting new structure
to extend galois analysis

and it's not just any transcendental extension

it corresponds to the exponential extensions
through the maps with the generalised trigonometry

so this could be a useful way to extract information
important to "big" conjectures
like that of schanuel

of course it might not give anything useful
(and i'm likely missing many stupid things)
but some study transcendence theory
and a couple of interesting leads
at least give a path for study

but ideas like trying to understand
even just it's abelian structure
seem like a task for many

but new question:

in what sense is this maximal?

is this maximal over algebraic extensions of Q?
or over all extensions of Q?

i've been pouring over kroenecker-weber
  which really isn't that long of a proof really
and i still haven't figured out
if it excludes some of the transcendental extensions
  built from the generalised polynomials
  that come from the generalised trigonometry

i.e.

are there abelian subfields of

  lim W[x] / (g)
  <--
   g

where W_n[x] is the ring of generalised polynomials
  of the form

   ---
   \       e_a
   /    a x
   ---
  a e A
 A finite
  A c C_n
e_a c C^_n

where C_n is a cyclotomic field
and C^_n it's ring of integers
(ie. as you've pointed out in the past
   the semigroup ring built from cyclotomics)

  ?

or are all such limits' center always F?

i've been able to find a lot of algebraic properties
  of these transcendental extensions
but i'm still very fuzzy
about this very foundational property...

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar

I (still) don't understand what precisely the generalised
polynomials in W_n[x] are.

To simplify notation a bit let K be a cyclotomic field and let
O be the ring of integers of K.

Sticking to the theory of semi-group rings (which my be the
wrong approach here) you seem to consider maps

f: O --> K

having finite support.
Such maps are added pointwise.
Multiplication however - if we follow the situation for polynomial
rings - should be defined using some sort of Cauchy product
rule. Can you explain how it works?

products work just like polynomial products

as an example
  look at the cyclotomic field of degree 3

example "polynomials" might be

     w
      3
x + x   + 1

and

        2
 w     w
  3     3
x   + x   - 1

where w_3 is the cube root of unity

multiplying these together gives
  term-by-term

            2                    2
 1+w     1+w         2w     w + w
    3       3          3     3   3
x     + x     - x + x    + x

                2
   w     w     w
    3     3     3
- x   + x   + x   - 1

which might be simplified to

          2                      2
 2w     -w     -w               w
   3      3      3    -1         3
x    + x    + x    + x   - x + x   - 1

Given W[x] is defined, what is g in the projective limit appearing
in your post?

g was meant to range over "irreducibles"
  (with a few ambiguities handwaved away)

perhaps a better characterisation
  of the generalised polynomial rings i am interested in
can be built on the definition starting with the form

K[x0, x1, ..., x_(n-1)]

and quotienting out terms like

  x0 x1 ... x_(n-1) = 1

(i.e. modding over the ideal (x0 x1 ... x_(n-1) - 1))

it's much easier to see the formal variables
  are all units
and similar properties from this form

then
taking this construction as the "polynomial" ring
  taking quotient rings with
    ideals generated from irreducible elements
should give something that looks like a number field extension

the point or motivation behind the construction
  where the projective limits were used
was to build extensions of the algebraic numbers
that still have many usable properties

or at least
  that's the hope

i'm still banging my head against them
  to try to figure out which properties are attackable
and i've made some basic characterisations
but i'm still not practiced enough in the tools

these rings come from the generalised trigonometry
where they are the rings that are used
  to invert the trigonometrics
(much as in the standard log forms
 of the classical trigonometrics)

these rings have tchebyshef maps on important elements
(and also horizontal maps
 and other tools from the trigonometry)

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, topposter
.

Relevant Pages

  • Re: field generated by the set of roots of unity
    ...   which really isn't that long of a proof really ... if it excludes some of the transcendental extensions ... polynomials in W_nare. ... Sticking to the theory of semi-group rings (which my be the ...
    (sci.math)
  • Re: understanding formal objects in computer languages
    ...   was using that as an illustration ... in the ring. ... we should freely replace this value ax with two polynomials ... many writers will skip the formality of the "marking of elements" ...
    (sci.math)
  • Re: solving the equation
    ...    to have to deal with? ... They are not necessarily linear functions. ... independent functions, which means that no linear combination ... An example set might be the polynomials: ...
    (sci.math)
  • Re: Building commuting polynomials...
    ... the first terms of the Chebyshev polynomials ... r positive real numbers,still gives commuting ...   just by definition ... which have their own generalised tchebyshef ...
    (sci.math)
  • Re: solving the equation
    ... Integral 0_1 fdx   ... using eqon integrating ... If you just have a maximum polynomial order that you want to deal ... Regarding handling large numbers of equations and polynomials ...
    (sci.math)