Download NOTE TO THEOREM 2.13.6. In the proof of Theorem 2.13.6

NOTE TO THEOREM 2.13.6.
IAN KIMING
In the proof of Theorem 2.13.6 (Hermite’s Discriminant Theorem) it is better to
define the set M slightly differently. Define M as the set of (λ1 , . . . , λn ) ∈ Rn such
that:

p
|λ1 |
< 2r1 −1 |DK | + 1





For r1 > 1 :
for i = 2, . . . , r1
|λi |
< 12 ,




 2
λj + λ2j+r2 < 1 ,
for j = r1 + 1, . . . , r1 + r2
For r1 = 0 :











|λ1 |
<
1
2
|λ1+r2 |
<
p
λ2j + λ2j+r2
|DK |
< 1,
for j = 2, . . . , r2 .
Then,
vol(M ) =
 r p
 2 1 |DK | + 1 · π r2
for r1 > 1
p
2 |DK | · π r2 −1
for r1 = 0 ,
p
whence vol(M ) > 2r1 +r2 |DK | in all cases. As
bounded,
centrally
pM is nownopen,
p
r1 +r2
−r2
symmetric, and convex with volume > 2
|DK | = 2 · 2
|DK | we deduce
from Minkowski’s lattice point theorem the existence of a lattice point α 6= 0 with
α ∈ M.

For the end of the argument where we show that α generates K, we may clearly
assume n > 1.
In both cases, i.e., r1 > 1 and r1 = 0, we have then |gj α| < 1 for j = 2, . . . , n, so
that we deduce |g1 α| > 1.
If r1 > 1 we then see that i = 1 is the only i with |gi α| > 1.
If r1 = 0 we have |gi α| > 1 precisely for i = 1 and for i = 1+r2 . Now α can not be
a real number since we would then obtain the contradiction 1 < |g1 α| = |Re(α)| < 12
(again because α ∈ M ). Hence Im(α) 6= 0, and so we see that there is exactly one
i such that |gi α| > 1 and Im(gi α) > 0 (remember that gi+r2 α is the complex
conjugate of gi α).
References
[1] H. Koch: ‘Number Theory. Algebraic Numbers and Functions’. Graduate Studies in Mathematics 24, AMS 2000.
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IAN KIMING
Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK2100 Copenhagen Ø, Denmark.
E-mail address: [email protected]