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Quadratic Number Fields – Lecture 6
TU Kaiserslautern – Summer term 2017
Tommy Hofmann
June 12, 2017
Lattices and J -sets
I If α, β ∈ C× are complex numbers with αβ 6∈ R, then
Λ = {aα + bβ | a, b ∈ Z}
is called a lattice and (α, β) is called a basis of Λ.
I A basis (α, β) of a lattice Λ ⊆ C is called normalized, ifIm(βα) > 0.
I The J -set of Λ is defined to be the set
J (Λ) =
{β
α
∣∣∣∣ (α, β) normalized basis
}.
I Λ ∼ Λ′ ⇔ J (Λ) = J (Λ′) ⇔ J (Λ) ∩ J (Λ′) 6= ∅.
The lattice Z[i ]
R
iR
−4
−4i
−3
−3i
−2
−2i
−1
−1i
1
1i
2
2i
3
3i
4
4i
J (Z[i ])100 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
J (Z[i ])500 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
J (Z[i ])1000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
J (Z[i ])5000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
J (Z[i ])10000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
The lattice 〈2, 1 +√−5〉
R
iR
−4
−4i
−3
−3i
−2
−2i
−1
−1i
1
1i
2
2i
3
3i
4
4i
J (〈2, 1 +√−5〉)
100 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
J (〈2, 1 +√−5〉)
500 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
J (〈2, 1 +√−5〉)
1000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
J (〈2, 1 +√−5〉)
5000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
J (〈2, 1 +√−5〉)
10000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1