NetNews Usenet Archive 1992 #27/NN_1992_27.iso/spool/sci/math/15397

Internet Message Format | 1992-11-23 | 2.9 KB

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From: [e-mail] (Richard Pinch)
Newsgroups: sci.math
Subject: Re: Help with Algebraic Number Theory
Keywords: hilbert algebraic number
Message-ID: <[e-mail]>
Date: 23 Nov 92 11:54:00 GMT
References: <[e-mail]>
Sender: [e-mail] (USENET news)
Organization: Department of Pure Mathematics, University of Cambridge
Lines: 55
Nntp-Posting-Host: emu.pmms.cam.ac.uk

In article <[e-mail]> [e-mail] writes:
>
>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.
>
We need to show that if  u \in U_{L'} is annihiliated by the
norm map N then some power of u, say u^m, is in V_L.  Now u = x + y sqrt(d)
with x, y \in L, and N(u) = x^2 - dy^2 = 1, so the only question
is about the integrality of x and y.  By the lemma cited, there is an m
such that u^m + u^{-m} is divisible by 2sqrt(d).
Let u^m = X + Y sqrt(d).  Now u^m and u^{-m} are algebraic integers,
so u^m+u^{-m} = 2X is: but 2X is divisible by 2sqrt(d) and so in particular
X is integral, and even divisible by sqrt(d).  Now Y sqrt(d) = u^m - x is
a difference of integers, each divisible by sqrt(d) and hence an integer
divisible by sqrt(d).  So X and Y are integers and so in O_L.
The finiteness of the index follows from applying this argument to a
set of generators u_i for U_L' and then taking the lcm of the resulting m_i.

Richard Pinch