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# N )/σ. The main purpose herein is to give a complete description of all the reduced ideals which are in the principal class of Q( √

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(1)

LXI.3 (1992)

by

(2)

0

1

π

0

i

i

i

0

0

i+1

i

i

i

i+1

i

i+12

i

0

i

i

i

i

0

1

2

π−1

0

0

1

2

π−1

0

i+1

π−i

i

π−i

i

π−i

i

0

i

0

1

2

k

j

j

j

k

π+k

k

k

k

k

0

i

i+1

i+2

i

0

i+1

0

(3)

i+2

0

i

0

i

0

r

i+1

0

s

i+2

0

t

j

0

r

j−1

0

s

j−2

0

t

i+1

r

i+1

r

n

k

i

0

i

0

3

1

i

n

k

2

2

n

(4)

−2

−1

−2

0

−1

0

0

−1

−1

1

0

1

1

0

0

2

1

2

2

1

i−2

i

i−1

i

i

i−1

m−2

m

m−1

m

m

m−1

m

m

m−1

m−1

m−2

−2

−1

−2

−1

n+1

n+1

n

n−1

n+1

n+1

n

n−1

k+1

k

k

m

m

k

m

m

j−3

j−2

j−3

j−2

−2

m

j−3

j−2

j−3

j−2

−1

j

j

j

j

j

j

j

j

j

j

j

j

j+1

j

j

j+1

j

j

(5)

j

j

j

j

j

j

j

j

j

j

j+1

j

j

j+1

j

j

j

j+1

j

j

j+1

j

j

j

j

j+1

j

j

j+1

i,j

σj

i,j

i

i,j

i,j

%j

i,j

i

i,j

−1,j

%j

−σj

−1,j

i,j

i,j

−1,j

i,j

i

i,j

i

i,j

i

−1,j

m,j

−1,j

m,j

m,j

m−1,j

m,j

m−1,j

m,j

m−1,j

m−1,j

σj

i+1,j

i+1,j

i,j

i−1,j

i−1,j

i+1,j

i,j

i+1,j

−2,j

−2,j

−1,j

%j

−1,j

0,j

%j

0,j

0,j

i+1,j

i+1,j

i,j

i,j

i,j

%j

(6)

m−2,j

σj

m−2,j

m−2,j

m−1,j

σj

m−1,j

m−1,j

σj

m−1,j

m,j

i−1,j

i+1,j

i,j

i,j

i,j

σj

i,j

i,j

i

−n

i

i

i

i

i

i

i

κ

κ

κ

κ

i=1

i

r

r

i=1

i

rj

j

j

j

rj+1

h

h

h

h

h

n

i−1,j

i,j

i−1,j

i,j

2

h

n

i,j

i,j

i,j

i,j

2

h+1

h

h

h

n

k

n

i,j

i+1,j

i,j

i+1,j

2

h+1

n

h+1

n

k

h+1

h

n

i+1,j

i+1,j

i+1,j

i+1,j

2

(7)

0

n

k

0

0

n

k

1

n

k

2

n

1

n

2

k

1

2

n−k

ψ(j)

n

k

ψ(j)

n−λj

ψ(j)

λj

j

λj

k−n+λj

j

j

j

k−n+λj

%j

−1,j

−n(bjk/nc+1)

j

j

ψ(j)+1

n

ψ(j)+1

k+λj

λj+1

ψ(j)+1

n−k−λj

n−λj+1

j

j

ψ(j)+i

n

i−3,j

i−2,j

i−3,j

i−2,j

2

ψ(j)+i

n

i−2,j

i−2,j

i−2,j

i−2,j

2

ψ(j)+i

i−1,j

2n−k−λj

n−λj

j

j

n−λj

σj

j

j

j

θ+1

n

k

θ+1

k

n−λj

θ+1

n−k

λj

θ+2

θ+2

n

θ+3

n

k

θ+2

n

θ+3

(8)

3

n

k

3

n−k

n−λ1

3

k

1

k

2k−n

1

1

1

−n

j

ψ(j)

ψ(j)

ψ(j)

ψ(j)+1

ψ(j)

ψ(j)

ψ(j)

n

k

n

ψ(j)+1

n

ψ(j)+1

n

k

ψ(j)+1

ψ(j)

λj+1

ψ(j)+1

ψ(j)+1

ψ(j)+1

n−λj+1

ψ(j+1)

n

k

ψ(j+1)

ψ(j)+2

ψ(j)+1

ψ(j)+1

ψ(j)+1

n

k

n

k

ψ(j+1)

n−λj+1

ψ(j+1)

ψ(j)+2

n

ψ(j)+2

n

k

ψ(j)+2

ψ(j)+1

n−λj+1

ψ(j+1)

λj+1

j+1

λj+1

k−n+λj+1

j+1

j+1

j

j

ψ(j)+1

ψ(j)

ψ(j)

ψ(j)

n

k

λj

k−n+λj

j

n−λj

n

k

n

k

n

k

j

n−λj

−2,j

−1,j

n

−2,j

−1,j

2

ψ(j)+1

ψ(j)

n

ψ(j)+1

n

k

ψ(j)+1

n

n

k

n−λj

j

n

k

n

k

j

n−λj

n

j

n−λj

n

k

n−λj

j

j

2n−λj

n−λj

j

k

n−λj

j

2

(9)

ψ(j)+1

n

j

σj

j

%j

j

2

n

−1,j

−1,j

−1,j

−1,j

2

ψ(j)+i

n

i−2,j

i−2,j

i−2,j

i−2,j

2

ψ(j)+i

i−3,j

i−2,j

n

i−3,j

i−2,j

2

j

κ

−n(rj+1)

j

rj+1

j

rj+1

j

rj+1

j

ψ(j)+m+2

n

m−1,j

m,j

n

m−1,j

m,j

2

ψ(j)+m+2

n

m,j

m,j

m,j

m,j

2

ψ(j)+m+2

n

σj

m−1,j

%j

2

n

k

%j

m−1,j

ψ(j)+m+2

2

%j

2

%j

λj+1

k−n+λj

m−1,j

σj

n−λj

j

m−1,j

j

ψ(j)+m+2

2n−k−λj

n−λj

j

ψ(j+1)

ψ(j)+m+3

2n−k−λj

n−λj

j

k−n+λj

n

k

j

k−n+λj

n

k

n

k

ψ(j+1)

ψ(j)+m+3

n

k−n+λj

2n−k−λj

n−λj+1

ψ(j+1)

ψ(j)+m+3

λj+1

j+1

λj+1

k−n+λj+1

j+1

j+1

(10)

j

θ+1

n

k

θ+1

k

n−λj

θ+1

n−k

λj

θ+2

θ+2

n

k

θ+3

n

k

θ+2

n

θ+3

k

θ+3

j

j

rj

k

6

2

6

1

1

1

1

1

1

1

1

1

0

n

k

0

0

0

n

k

1

n

k

1

2

n

2

1

n

2

k

1

2

n−k

ψ(1)

3

n

k

3

ψ(1)

3

n−λ1

ψ(1)

3

λ1

k−n+λ1

1

(11)

ψ(1)+1

4

n

−2,1

−1,1

−2,1

−1,1

2

6

4

ψ(1)+1

4

n

−1,1

−1,1

−1,1

−1,1

2

6

ψ(1)+1

4

0

ψ(1)+2

5

n

−1,1

0,1

−1,1

0,1

2

6

5

ψ(1)+2

5

n

0,1

0,1

0,1

0,1

2

6

ψ(1)+2

5

2n−k−λ1

n−λ1

1

4

2

θ+1

6

n

k

6

4

6

6

k

6

n

k

7

7

n

7

8

n

k

8

7

n

8

i

i

i

i

i

ψ(j)

n

k

ψ(j)

n−λj

(12)

ψ(j)

λj

j

λj

k−n+λj

j

j

ψ(j)+1

n

ψ(j)+1

λj+k

ψ(j)+1

n−λj−k

j

ψ(j)+1

n

k

n−λj

ψ(j)+1

n

k

n−λj

k−n+λj

ψ(j)+1

ψ(j)+2

n

k

k−n+λj

ψ(j)+2

k−n+λj

ψ(j)+2

2n−k−λj

n−λj

ψ(j)+3

ψ(j+1)

n

k

ψ(j+1)

2n−k−λj

n−λj+1

ψ(j+1)

λj+1

j+1

λj+1

k−n+λj+1

j+1

0j

j0

νj

0j

j

j0

m,j

m−1,j

σj

m−1,j

m−1,j

σj

m−1,j

m−1,j

σj

νj+1

−1,j

−n−νj

%j

−1,j

m−2,j

m,j

m,j

m−1,j

m−2,j

m,j

νj+1

j+10

m,j

m,j

m−1,j

m−2,j

m−2,j

m−1,j

m,j

m,j

m−1,j

m,j

j+1

m−2,j

j+1

νj+1

0j+1

m−1,j

νj+1

j+10

m−1,j

k−νj+1

0j+1

m−1,j

m−1,j

j+10

m−2,j

j+1

n

−2,j

−2,j

−2,j

−2,j

2

σj

(13)

α

α+1

j

j

j

j

k−n

2α+1

n

k

n

k

j

j

0j

j

j−1

j−1

j

−n−νj

j

0

j

i=1

0i

0

0

0j+1

0

j

i=1

i

j

i=1

i−1

j

j

j

i=1

i−1

j

i=1

i−1

j

i=1

j

i=1

i

r0j

0j

0

r0

j

j

0

n

k

n

k

2

2

n

ψ0(j)

j

j0

n

j0

k−νj

j

νj

j

j0

2

ψ0(j)

n

j0

νj

j0

k−νj

j0

2

ψ0(j)

j

k−n−νj

j

j

j

j

j

(14)

j

ψ0(j)+1

n

k

j0

k−νj

ψ0(j)+1

k−νj

ψ0(j)+1

νj+n−k

νj

0j

ψ0(j)+2

n

k

ψ0(j)+2

n−k−νj

−2,j

−2,j

n

−2,j

−2,j

2

ψ0(j)+2

k−νj

2k−n−νj

j

ψ0(j)+2+i

n

i−3,j

i−2,j

i−3,j

i−2,j

2

ψ0(j)+2+i

n

i−2,j

i−2,j

i−2,j

i−2,j

2

ψ0(j)+2+i

i−1,j

j

j

j

j

ψ0(j)+1

n

k

0j

k−νj

ψ0(j)+1

k−νj

ψ0(j)+1

νj

0j

ψ0(j)+2

k

ψ0(j)+2

n+νj

n

−2,j

−2,j

−2,j

−2,j

2

ψ0(j)+2

k−n−νj

j

ψ0(j)+2+i

n

i−3,j

i−2,j

i−3,j

i−2,j

2

ψ0(j)+2+i

n

i−2,j

i−2,j

i−2,j

i−2,j

2

ψ0(j)+2+i

i−1,j

j

j

ψ0(j)+1

n

k

0j

k−νj

ψ0(j)+1

k−νj

−1,j

−1,j

n

−1,j

−1,j

2

ψ0(j)+1

0,j

νj

j0

ψ0(j)+i

n

i−3,j

i−2,j

i−3,j

i−2,j

2

ψ0(j)+i

n

i−2,j

i−2,j

i−2,j

i−2,j

2

ψ0(j)+i

i−1,j

j

j

j

ψ0(j)+i

n

i−3,j

i−2,j

i−3,j

i−2,j

2

ψ0(j)+i

n

i−2,j

i−2,j

i−2,j

i−2,j

2

(15)

ψ0(j)+i

i−1,j

0

j

h

0

λ+1

j

j0

n

j0

k−νj

j

νj

j

j0

2

λ+1

n

j0

νj

0j

k−νj

j0

2

λ+1

j

j

λ+2

n

k

0j

k−νj

λ+3

n

k

λ+2

n

λ+3

λ+2

k−n

j

0

0j

0

κ

κ

κ

0

j

j

j

j

j

j

4

2

4

0

1

2

3

0

1

2

3

k−ν0

k−ν1

k−ν2

k−ν3

0

1

2

3

0

1

2

3

4

00

01

02

30

04

i

0

i

0

0

0

0

0

(16)

ψ0(0)

1

4

ψ0(1)

4

8

ψ0(2)

9

7

ψ0(3)

14

6

j

ψ0(0)+1

2

5

ψ0(0)+2

3

ψ0(1)+1

5

j

ψ0(1)+2

6

3

ψ0(1)+3

7

6

ψ0(2)+1

10

2

ψ0(2)+2

11

2

ψ0(2)+3

12

7

ψ0(3)+1

15

3

ψ0(3)+2

ψ0(3)+3

17

8

ψ0(4)

19

5

ψ0(4)+1

20

4

ψ0(4)+2

21

i

i

i

i

i

i

i

i

i

4

9

14

i

0

### We need a preliminary result.

Cytaty

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