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Quadratic Number Fields – Lecture 6
TU Kaiserslautern – Summer term 2017
Tommy Hofmann
June 12, 2017
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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= ∅.
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The lattice Z[i ]
R
iR
−4
−4i
−3
−3i
−2
−2i
−1
−1i
1
1i
2
2i
3
3i
4
4i
![Page 4: Quadratic Number Fields { Lecture 6 · Quadratic Number Fields { Lecture 6 TU Kaiserslautern { Summer term 2017 Tommy Hofmann June 12, 2017. Lattices and J-sets I If ; 2C are complex](https://reader036.vdocuments.net/reader036/viewer/2022063015/5fd2fd419131bc4b284ffd13/html5/thumbnails/4.jpg)
J (Z[i ])100 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
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J (Z[i ])500 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
![Page 6: Quadratic Number Fields { Lecture 6 · Quadratic Number Fields { Lecture 6 TU Kaiserslautern { Summer term 2017 Tommy Hofmann June 12, 2017. Lattices and J-sets I If ; 2C are complex](https://reader036.vdocuments.net/reader036/viewer/2022063015/5fd2fd419131bc4b284ffd13/html5/thumbnails/6.jpg)
J (Z[i ])1000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
![Page 7: Quadratic Number Fields { Lecture 6 · Quadratic Number Fields { Lecture 6 TU Kaiserslautern { Summer term 2017 Tommy Hofmann June 12, 2017. Lattices and J-sets I If ; 2C are complex](https://reader036.vdocuments.net/reader036/viewer/2022063015/5fd2fd419131bc4b284ffd13/html5/thumbnails/7.jpg)
J (Z[i ])5000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
![Page 8: Quadratic Number Fields { Lecture 6 · Quadratic Number Fields { Lecture 6 TU Kaiserslautern { Summer term 2017 Tommy Hofmann June 12, 2017. Lattices and J-sets I If ; 2C are complex](https://reader036.vdocuments.net/reader036/viewer/2022063015/5fd2fd419131bc4b284ffd13/html5/thumbnails/8.jpg)
J (Z[i ])10000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
![Page 9: Quadratic Number Fields { Lecture 6 · Quadratic Number Fields { Lecture 6 TU Kaiserslautern { Summer term 2017 Tommy Hofmann June 12, 2017. Lattices and J-sets I If ; 2C are complex](https://reader036.vdocuments.net/reader036/viewer/2022063015/5fd2fd419131bc4b284ffd13/html5/thumbnails/9.jpg)
The lattice 〈2, 1 +√−5〉
R
iR
−4
−4i
−3
−3i
−2
−2i
−1
−1i
1
1i
2
2i
3
3i
4
4i
![Page 10: Quadratic Number Fields { Lecture 6 · Quadratic Number Fields { Lecture 6 TU Kaiserslautern { Summer term 2017 Tommy Hofmann June 12, 2017. Lattices and J-sets I If ; 2C are complex](https://reader036.vdocuments.net/reader036/viewer/2022063015/5fd2fd419131bc4b284ffd13/html5/thumbnails/10.jpg)
J (〈2, 1 +√−5〉)
100 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
![Page 11: Quadratic Number Fields { Lecture 6 · Quadratic Number Fields { Lecture 6 TU Kaiserslautern { Summer term 2017 Tommy Hofmann June 12, 2017. Lattices and J-sets I If ; 2C are complex](https://reader036.vdocuments.net/reader036/viewer/2022063015/5fd2fd419131bc4b284ffd13/html5/thumbnails/11.jpg)
J (〈2, 1 +√−5〉)
500 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
![Page 12: Quadratic Number Fields { Lecture 6 · Quadratic Number Fields { Lecture 6 TU Kaiserslautern { Summer term 2017 Tommy Hofmann June 12, 2017. Lattices and J-sets I If ; 2C are complex](https://reader036.vdocuments.net/reader036/viewer/2022063015/5fd2fd419131bc4b284ffd13/html5/thumbnails/12.jpg)
J (〈2, 1 +√−5〉)
1000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
![Page 13: Quadratic Number Fields { Lecture 6 · Quadratic Number Fields { Lecture 6 TU Kaiserslautern { Summer term 2017 Tommy Hofmann June 12, 2017. Lattices and J-sets I If ; 2C are complex](https://reader036.vdocuments.net/reader036/viewer/2022063015/5fd2fd419131bc4b284ffd13/html5/thumbnails/13.jpg)
J (〈2, 1 +√−5〉)
5000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1
![Page 14: Quadratic Number Fields { Lecture 6 · Quadratic Number Fields { Lecture 6 TU Kaiserslautern { Summer term 2017 Tommy Hofmann June 12, 2017. Lattices and J-sets I If ; 2C are complex](https://reader036.vdocuments.net/reader036/viewer/2022063015/5fd2fd419131bc4b284ffd13/html5/thumbnails/14.jpg)
J (〈2, 1 +√−5〉)
10000 points
R
iR
−2 −1 0 1 2
0:25
0:5
0:75
1