NetNews Usenet Archive 1992 #27/NN_1992_27.iso/spool/sci/math/15391

Internet Message Format | 1992-11-22 | 2.9 KB

Path: sparky!uunet!munnari.oz.au!manuel.anu.edu.au!mehta!molrchon
From: [e-mail] (Rory Molinari )
Newsgroups: sci.math
Subject: Help with Algebraic Number Theory
Date: 23 Nov 1992 03:35:28 GMT
Organization: Australian National University
Lines: 69
Distribution: world
Message-ID: <[e-mail]>
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Keywords: hilbert algebraic number

Hi,
    I'm a maths undergraduate, nearing the end of my 4th (and last) year.
As a result, the due-date on my Honours thesis is looming, and I am still stuck
with a couple of problems that neither I nor my project supervisor have
been able to crack.  I am hoping someone on the net might be able to give
me a few hints.

My thesis topic is "Hilbert's 10th Problem and a Generalization to
Algebraic Integers".  In the course of the year, one of the papers I read
was:

    Denef & Lipshitz, "Diophantine Sets Over Some Rings of Algebraic Integers",
    J. London Math. Soc., v18, 1978, pp385-391.

and I am covering the contents of that paper in my thesis.  One of the theorems
proved in that paper is the following:

Notation:  For a number field L (ie. a field of algebraic numbers, of finite
  degree over the rationals) the ring of algebraic integers in L will be
  denoted O(L).  The multiplicative group of units in O(L) will be denoted U_L.
  The torsion free rank of a group G will be denoted rk G.

THEOREM:
  Let L be any number field and d \in O(L) such that sqrt(d) \not\in L.
  Write L' = L(sqrt(d)).  Define

       V_L = {x + y.sqrt(d) : x,y \in O(L) and x^2 - dy^2 = 1}

  Then V_L is a subgroup of U_{L'} and

       rk V_L = rk U_{L'} - rk U_L
ENDTHEOREM

The proof suggested by Denef & Lipshitz starts as follows:

  Let N:U_{L'} -> U_L be the restriction of the norm from L' to L.  In this case
  we have N(u + v.sqrt(d)) = u^2 - dv^2, so that V_L is a subgroup of
  kernel(N).  The claim is that V_L in fact has finite index in kernel(N).
  Denef & Lipshitz do not explicitly prove this, but say it follows easily from
  the following lemma:

    Suppose K is a number field, e \in U_K and 0 != t \in O(K).  Then there is
    m > 0 such that t | e^m - e^{-m}.

  My supervisor and I have been unable to see quite how this implies what we need.



The second problem I have is much more minor, but I can't for the life of me
see how it.  Suppose J is some number field, and x \in O(J).  Then the claim
is that:

      x != 0  <==>  (\exist v, y \in O(J))[xy = (2v-1)(3v-1)]

It apparently follows from the Chinese Remainder Theorem, but I can't see the
trick.  No doubt it is embarassingly easy.




   If anybody has any ideas I would be very grateful.  Full credit will be given
in my thesis (though I don't suppose this is  much of a reward :).
I don't think this has much global appeal, so replies by email are probably
best.


Cheers,
Rory Molinari
[e-mail]