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Examples of Kummer extensions over the rational function field

by Y. Lee and J. Yoo.

We list some Kummer extensions K over k = F

q

(T) of extension degree ` whose divisor class numbers are not divisible by the extension degree `. Furthermore, we give some examples of Kummer extensions K

n

= k( p

`

P

n

(T )) whose divisor class numbers are divisible by `, where P

n

(T) = P

0

(T

mn

), α is a root of P

0

(T ) ∈ F

q

, Q = q

degP0

, and m is a prime which satisfies the followings:

(i) α 6∈ ( F

q

)

m

; (ii) m | Q − 1.

We use the following notation.

Notation

q a prime power

` a prime divisor of q − 1

P

i

= P

i

(T ) monic irreducible polynomial in F

q

[T ] for every i Q(T ) aP

1

P

2

· · · P

t

, where a ∈ F

∗q

k = F

q

(T ) the rational function field K = k( p

`

Q(T )) a Kummer extension of extension degree `

t the number of finite preims of k which are ramified in K

∞ the infinite prime of k

g the genus of K

d

i

the degree of P

i

(T ) for every i with 1 ≤ i ≤ t

δ the degree of Q(T )

h

K

the divisor class number of K

k(Λ

P

) the P th cyclotomic function field

k(Λ

P

)

+

the maximal real subfield of k(Λ

P

)

(2)

(i) ∞ is totally ramified

Table 1. Divisor class numbers of Kummer extensions with ` = 2

q δ Q(T ) g h

K

3 3

T

3

+ 2T + 2

1

1 ≡ 1 (mod 2) T

3

+ T

2

+ 2

3 ≡ 1 (mod 2) T

3

+ T

2

+ T + 2

T

3

+ T

2

+ 2T + 1

5 ≡ 1 (mod 2) T

3

+ 2T

2

+ 1

T

3

+ 2T

2

+ T + 1 T

3

+ 2T

2

+ 2T + 2

T

3

+ 2T + 1 7 ≡ 1 (mod 2)

5 5

T

5

+ 4T + 1

2

71

T

5

+ 4T + 2 11 ≡ 1 (mod 2)

T

5

+ 4T + 3 11 ≡ 1 (mod 2)

T

5

+ 4T + 4 71 ≡ 1 (mod 2)

T

5

+ T

2

+ 2 25 ≡ 1 (mod 2)

T

5

+ T

2

+ T + 4 25 ≡ 1 (mod 2)

T

5

+ T

2

+ 2T + 4 25 ≡ 1 (mod 2)

T

5

+ T

2

+ 3T + 2 25 ≡ 1 (mod 2)

T

5

+ T

2

+ 4T + 3 25 ≡ 1 (mod 2)

T

5

+ 2T

2

+ 1 7 ≡ 1 (mod 2)

T

5

+ 2T

2

+ T + 2 37 ≡ 1 (mod 2)

T

5

+ 2T

2

+ 2T + 2 37 ≡ 1 (mod 2)

T

5

+ 2T

2

+ 3T + 1 37 ≡ 1 (mod 2)

T

5

+ 2T

2

+ 4T + 4 37 ≡ 1 (mod 2)

T

5

+ 3T

2

+ 4 37 ≡ 1 (mod 2)

T

5

+ 3T

2

+ T + 3 37 ≡ 1 (mod 2)

T

5

+ 3T

2

+ 2T + 3 37 ≡ 1 (mod 2)

T

5

+ 3T

2

+ 3T + 4 37 ≡ 1 (mod 2)

T

5

+ 3T

2

+ 4T + 1 37 ≡ 1 (mod 2)

T

5

+ 4T

2

+ 3 25 ≡ 1 (mod 2)

T

5

+ 4T

2

+ T + 1 25 ≡ 1 (mod 2)

T

5

+ 4T

2

+ 2T + 1 25 ≡ 1 (mod 2)

T

5

+ 4T

2

+ 3T + 3 25 ≡ 1 (mod 2)

T

5

+ 4T

2

+ 4T + 2 25 ≡ 1 (mod 2)

T

5

+ T

3

+ 2T + 2 23 ≡ 1 (mod 2)

T

5

+ T

3

+ 2T + 3 23 ≡ 1 (mod 2)

T

5

+ T

3

+ 3T + 2 15 ≡ 1 (mod 2)

T

5

+ T

3

+ 3T + 3 15 ≡ 1 (mod 2)

T

5

+ T

3

+ T

2

+ 3 15 ≡ 1 (mod 2)

T

5

+ T

3

+ T

2

+ 4 15 ≡ 1 (mod 2)

T

5

+ T

3

+ T

2

+ 4T + 1 23 ≡ 1 (mod 2)

T

5

+ T

3

+ T

2

+ 4T + 2 23 ≡ 1 (mod 2)

T

5

+ T

3

+ 2T

2

+ 3 23 ≡ 1 (mod 2)

T

5

+ T

3

+ 2T

2

+ 4 23 ≡ 1 (mod 2)

T

5

+ T

3

+ 2T

2

+ T + 2 15 ≡ 1 (mod 2)

T

5

+ T

3

+ 2T

2

+ T + 3 15 ≡ 1 (mod 2)

T

5

+ T

3

+ 3T

2

+ 1 23 ≡ 1 (mod 2)

T

5

+ T

3

+ 3T

2

+ 2 23 ≡ 1 (mod 2)

T

5

+ T

3

+ 3T

2

+ T + 2 15 ≡ 1 (mod 2)

(3)

q δ Q(T ) g h

K

T

5

+ T

3

+ 3T

2

+ T + 3 15 ≡ 1 (mod 2)

5 5

T

5

+ T

3

+ 4T

2

+ 1

2

15 ≡ 1 (mod 2)

T

5

+ T

3

+ 4T

2

+ 2 15 ≡ 1 (mod 2)

T

5

+ T

3

+ 4T

2

+ 4T + 3 23 ≡ 1 (mod 2)

T

5

+ T

3

+ 4T

2

+ 4T + 4 23 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 1 37 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 4 37 ≡ 1 (mod 2)

T

5

+ 2T

3

+ T + 2 19 ≡ 1 (mod 2)

T

5

+ 2T

3

+ T + 3 19 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 2T + 1 51 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 2T + 4 51 ≡ 1 (mod 2)

T

5

+ 2T

3

+ T

2

+ T + 2 37 ≡ 1 (mod 2)

T

5

+ 2T

3

+ T

2

+ T + 4 37 ≡ 1 (mod 2)

T

5

+ 2T

3

+ T

2

+ 2T + 1 19 ≡ 1 (mod 2)

T

5

+ 2T

3

+ T

2

+ 2T + 2 19 ≡ 1 (mod 2)

T

5

+ 2T

3

+ T

2

+ 3T + 1 51 ≡ 1 (mod 2)

T

5

+ 2T

3

+ T

2

+ 3T + 4 51 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 2T

2

+ 2 19 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 2T

2

+ 3 19 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 2T

2

+ T + 1 51 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 2T

2

+ T + 3 51 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 2T

2

+ 4T + 2 37 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 2T

2

+ 4T + 4 37 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 3T

2

+ 2 19 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 3T

2

+ 3 19 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 3T

2

+ T + 2 51 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 3T

2

+ T + 4 51 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 3T

2

+ 4T + 1 37 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 3T

2

+ 4T + 3 37 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 4T

2

+ T + 1 37 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 4T

2

+ T + 3 37 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 4T

2

+ 2T + 3 19 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 4T

2

+ 2T + 4 19 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 4T

2

+ 3T + 1 51 ≡ 1 (mod 2)

T

5

+ 2T

3

+ 4T

2

+ 3T + 4 51 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 2 25 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 3 25 ≡ 1 (mod 2)

T

5

+ 3T

3

+ T + 1 55 ≡ 1 (mod 2)

T

5

+ 3T

3

+ T + 4 55 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 2T + 2 15 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 2T + 3 15 ≡ 1 (mod 2)

(4)

q δ Q(T ) g h

K

5 5

T

5

+ 3T

3

+ 2T

2

+ 3T + 3

2

15 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 3T

2

+ T + 3 25 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 3T

2

+ T + 4 25 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 3T

2

+ 2T + 2 55 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 3T

2

+ 2T + 4 55 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 3T

2

+ 3T + 2 15 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 3T

2

+ 3T + 3 15 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 4T

2

+ 1 55 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 4T

2

+ 4 55 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 4T

2

+ T + 3 15 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 4T

2

+ T + 4 15 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 4T

2

+ 4T + 1 25 ≡ 1 (mod 2)

T

5

+ 3T

3

+ 4T

2

+ 4T + 2 25 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 2T + 1 35 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 2T + 4 35 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 3T + 1 51 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 3T + 4 51 ≡ 1 (mod 2)

T

5

+ 4T

3

+ T

2

+ 1 35 ≡ 1 (mod 2)

T

5

+ 4T

3

+ T

2

+ 3 35 ≡ 1 (mod 2)

T

5

+ 4T

3

+ T

2

+ T + 1 51 ≡ 1 (mod 2)

T

5

+ 4T

3

+ T

2

+ T + 4 51 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 2T

2

+ 2 51 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 2T

2

+ 4 51 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 2T

2

+ 4T + 1 35 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 2T

2

+ 4T + 3 35 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 3T

2

+ 1 51 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 3T

2

+ 3 51 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 3T

2

+ 4T + 2 35 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 3T

2

+ 4T + 4 35 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 4T

2

+ 2 35 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 4T

2

+ 4 35 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 4T

2

+ T + 1 51 ≡ 1 (mod 2)

T

5

+ 4T

3

+ 4T

2

+ T + 4 51 ≡ 1 (mod 2)

T

5

+ T

4

+ 4 35 ≡ 1 (mod 2)

T

5

+ T

4

+ T + 4 35 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T + 1 15 ≡ 1 (mod 2)

T

5

+ T

4

+ T

2

+ 2T + 3 17 ≡ 1 (mod 2)

T

5

+ T

4

+ T

2

+ 3T + 1 33 ≡ 1 (mod 2)

T

5

+ T

4

+ T

2

+ 4T + 2 29 ≡ 1 (mod 2)

T

5

+ T

4

+ T

2

+ 4T + 4 69 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

2

+ 2 35 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

2

+ T + 3 27 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

2

+ 3T + 2 31 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

2

+ 4T + 3 23 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

2

+ 4T + 4 13 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

2

+ 1 55 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

2

+ 3 15 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

2

+ 2T + 2 31 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

2

+ 3T + 1 49 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

2

+ 3T + 3 29 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

2

+ 4T + 3 27 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

2

+ 3 15 ≡ 1 (mod 2)

(5)

q δ Q(T ) g h

K

5 5

T

5

+ T

4

+ 4T

2

+ T + 1

2

39 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

2

+ 2T + 1 33 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

2

+ 3T + 2 17 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

2

+ 4T + 1 31 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

2

+ 4T + 2 21 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

2

+ 4T + 4 71 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 3 33 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ T + 3 17 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 2T + 1 31 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 2T + 2 21 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 2T + 4 71 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 3T + 2 15 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 4T + 4 39 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ T

2

+ 2 15 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ T

2

+ T + 4 35 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ T

2

+ 2T + 3 35 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 2T

2

+ T + 2 17 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 2T

2

+ 2T + 4 33 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 2T

2

+ 3T + 1 69 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 2T

2

+ 3T + 4 29 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 3T

2

+ 1 31 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 3T

2

+ T + 1 23 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 3T

2

+ T + 2 13 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 3T

2

+ 2T + 4 35 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 3T

2

+ 3T + 4 27 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 4T

2

+ 1 15 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 4T

2

+ 4 55 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 4T

2

+ 2T + 3 31 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 4T

2

+ 3T + 1 49 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 4T

2

+ 3T + 3 29 ≡ 1 (mod 2)

T

5

+ T

4

+ T

3

+ 4T

2

+ 4T + 2 27 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ T + 2 17 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ 2T + 3 33 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ 3T + 2 29 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ 3T + 4 69 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ T

2

+ 4 35 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ T

2

+ T + 3 27 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ T

2

+ 3T + 3 31 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ T

2

+ 4T + 2 23 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ T

2

+ 4T + 3 13 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ 2T

2

+ 1 27 ≡ 1 (mod 2)

(6)

q δ Q(T ) g h

K

5 5

T

5

+ T

4

+ 2T

3

+ 3T

2

+ 4T + 2

2

33 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ 4T

2

+ 2T + 2 15 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ 4T

2

+ 3T + 3 35 ≡ 1 (mod 2)

T

5

+ T

4

+ 2T

3

+ 4T

2

+ 4T + 1 35 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 2 29 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 4 69 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 3T + 4 17 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 4T + 4 33 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ T

2

+ 3 35 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ T

2

+ T + 1 27 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ T

2

+ 3T + 4 31 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ T

2

+ 4T + 2 23 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ T

2

+ 4T + 3 13 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 2T

2

+ T + 1 31 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 2T

2

+ 2T + 2 49 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 2T

2

+ 2T + 4 29 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 2T

2

+ 3T + 1 27 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 2T

2

+ 4T + 1 55 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 2T

2

+ 4T + 3 15 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 3T

2

+ 4 33 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 3T

2

+ T + 2 17 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 3T

2

+ 2T + 1 71 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 3T

2

+ 2T + 3 31 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 3T

2

+ 2T + 4 21 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 3T

2

+ 3T + 2 15 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 3T

2

+ 4T + 2 39 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 4T

2

+ T + 2 15 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 4T

2

+ 2T + 2 35 ≡ 1 (mod 2)

T

5

+ T

4

+ 3T

3

+ 4T

2

+ 3T + 4 35 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 1 17 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ T + 1 31 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ T + 2 21 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ T + 4 71 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 2T + 4 15 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 3T + 3 39 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 4T + 4 33 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ T

2

+ 2 35 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ T

2

+ 3T + 2 15 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ T

2

+ 4T + 1 35 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 2T

2

+ 1 69 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 2T

2

+ 4 29 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 2T

2

+ 3T + 3 17 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 2T

2

+ 4T + 2 33 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 3T

2

+ T + 4 31 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 3T

2

+ 2T + 1 23 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 3T

2

+ 2T + 2 13 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 3T

2

+ 3T + 1 35 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 3T

2

+ 4T + 3 27 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 4T

2

+ 1 55 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 4T

2

+ 3 15 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 4T

2

+ 2T + 4 31 ≡ 1 (mod 2)

T

5

+ T

4

+ 4T

3

+ 4T

2

+ 3T + 1 29 ≡ 1 (mod 2)

(7)

q δ Q(T ) g h

K

5 5

T

5

+ T

4

+ 4T

3

+ 4T

2

+ 3T + 4

2

49 ≡ 1 (mod 2) T

5

+ T

4

+ 4T

3

+ 4T

2

+ 4T + 2 27 ≡ 1 (mod 2) T

5

+ 2T

4

+ 3 23 ≡ 1 (mod 2) T

5

+ 2T

4

+ T + 3 23 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T + 2 51 ≡ 1 (mod 2) T

5

+ 2T

4

+ T

2

+ 4 23 ≡ 1 (mod 2) T

5

+ 2T

4

+ T

2

+ T + 1 39 ≡ 1 (mod 2) T

5

+ 2T

4

+ T

2

+ 3T + 4 19 ≡ 1 (mod 2) T

5

+ 2T

4

+ T

2

+ 4T + 1 35 ≡ 1 (mod 2) T

5

+ 2T

4

+ T

2

+ 4T + 3 49 ≡ 1 (mod 2) T

5

+ 2T

4

+ 2T

2

+ 1 51 ≡ 1 (mod 2) T

5

+ 2T

4

+ 2T

2

+ T + 2 27 ≡ 1 (mod 2) T

5

+ 2T

4

+ 2T

2

+ 2T + 2 21 ≡ 1 (mod 2) T

5

+ 2T

4

+ 2T

2

+ 3T + 4 29 ≡ 1 (mod 2) T

5

+ 2T

4

+ 2T

2

+ 4T + 2 19 ≡ 1 (mod 2) T

5

+ 2T

4

+ 2T

2

+ 4T + 3 11 ≡ 1 (mod 2) T

5

+ 2T

4

+ 2T

2

+ 4T + 4 33 ≡ 1 (mod 2) T

5

+ 2T

4

+ 3T

2

+ 2T + 1 53 ≡ 1 (mod 2) T

5

+ 2T

4

+ 3T

2

+ 3T + 2 21 ≡ 1 (mod 2) T

5

+ 2T

4

+ 3T

2

+ 4T + 3 9 ≡ 1 (mod 2) T

5

+ 2T

4

+ 3T

2

+ 4T + 4 41 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 4T

2

+ 1 27 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 4T

2

+ 2 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 4T

2

+ 2T + 4 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 4T

2

+ 3T + 1 41 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 4T

2

+ 3T + 2 13 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 4T

2

+ 4T + 1 39 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ 2 29 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ T + 2 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ T + 3 11 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ T + 4 33 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ 2T + 3 51 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ 3T + 1 27 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ 4T + 3 21 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ T

2

+ 2 9 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ T

2

+ 3 41 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ T

2

+ 3T + 1 53 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ T

2

+ 4T + 4 21 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ 2T

2

+ 1 27 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ 2T

2

+ 2 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ T

3

+ 2T

2

+ 2T + 3 19 ≡ 1 (mod 2)

(8)

q δ Q(T ) g h

K

5 5

T

5

+ 2T

4

+ 2T

3

+ 3

2

9 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 4 41 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 3T + 3 53 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 4T + 3 21 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ T

2

+ T + 2 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ T

2

+ 2T + 3 41 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ T

2

+ 2T + 4 13 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ T

2

+ 3T + 2 39 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ T

2

+ 4T + 1 27 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ T

2

+ 4T + 2 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 2T

2

+ T + 4 51 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 2T

2

+ 2T + 4 23 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 2T

2

+ 3T + 3 23 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 3T

2

+ 1 23 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 3T

2

+ T + 2 39 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 3T

2

+ 3T + 3 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 3T

2

+ 4T + 1 49 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 3T

2

+ 4T + 4 35 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 4T

2

+ 3 21 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 4T

2

+ T + 4 29 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 4T

2

+ 2T + 1 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 4T

2

+ 2T + 2 11 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 4T

2

+ 2T + 3 33 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 4T

2

+ 3T + 4 51 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 2T

3

+ 4T

2

+ 4T + 4 27 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ T + 4 53 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 2T + 1 21 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 3T + 3 9 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 3T + 4 41 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ T

2

+ 2 39 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ T

2

+ T + 3 27 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ T

2

+ T + 4 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ T

2

+ 3T + 3 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ T

2

+ 4T + 1 41 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ T

2

+ 4T + 2 13 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 2T

2

+ 2T + 4 51 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 2T

2

+ 3T + 1 23 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 2T

2

+ 4T + 2 23 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 3T

2

+ 3 23 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 3T

2

+ T + 1 39 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 3T

2

+ 3T + 1 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 3T

2

+ 4T + 1 49 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 3T

2

+ 4T + 4 35 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 4T

2

+ 2 29 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 4T

2

+ T + 1 19 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 4T

2

+ T + 2 11 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 4T

2

+ T + 3 33 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 4T

2

+ 2T + 1 51 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 4T

2

+ 3T + 3 27 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 3T

3

+ 4T

2

+ 4T + 4 21 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 4T

3

+ 1 21 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 4T

3

+ T + 1 29 ≡ 1 (mod 2)

(9)

q δ Q(T ) g h

K

T

5

+ 2T

4

+ 4T

3

+ 2T + 2 19 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 2T + 3 11 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 2T + 4 33 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 3T + 4 51 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 4T + 3 27 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ T

2

+ T + 4 53 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ T

2

+ 2T + 3 21 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ T

2

+ 3T + 2 9 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ T

2

+ 3T + 3 41 ≡ 1 (mod 2)

T

5

+ 2T

4

+ 4T

3

+ 2T

2

+ 2 27 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 2T

2

+ 3 19 ≡ 1 (mod 2)

5 5

T

5

+ 2T

4

+ 4T

3

+ 2T

2

+ 2T + 1

2

19 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 2T

2

+ 3T + 1 41 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 2T

2

+ 3T + 2 13 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 2T

2

+ 4T + 4 39 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 3T

2

+ 4 51 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 3T

2

+ T + 3 23 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 3T

2

+ 2T + 1 23 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 4T

2

+ 2 19 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 4T

2

+ T + 2 35 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 4T

2

+ T + 4 49 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 4T

2

+ 2T + 3 23 ≡ 1 (mod 2) T

5

+ 2T

4

+ 4T

3

+ 4T

2

+ 3T + 3 39 ≡ 1 (mod 2) T

5

+ 3T

4

+ 2 23 ≡ 1 (mod 2) T

5

+ 3T

4

+ T + 2 23 ≡ 1 (mod 2) T

5

+ 3T

4

+ 4T + 3 51 ≡ 1 (mod 2) T

5

+ 3T

4

+ T

2

+ 3 19 ≡ 1 (mod 2) T

5

+ 3T

4

+ T

2

+ 4 27 ≡ 1 (mod 2) T

5

+ 3T

4

+ T

2

+ 2T + 1 19 ≡ 1 (mod 2) T

5

+ 3T

4

+ T

2

+ 3T + 3 13 ≡ 1 (mod 2) T

5

+ 3T

4

+ T

2

+ 3T + 4 41 ≡ 1 (mod 2) T

5

+ 3T

4

+ T

2

+ 4T + 4 39 ≡ 1 (mod 2) T

5

+ 3T

4

+ 2T

2

+ 2T + 4 53 ≡ 1 (mod 2) T

5

+ 3T

4

+ 2T

2

+ 3T + 3 21 ≡ 1 (mod 2) T

5

+ 3T

4

+ 2T

2

+ 4T + 1 41 ≡ 1 (mod 2) T

5

+ 3T

4

+ 2T

2

+ 4T + 2 9 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

2

+ 4 51 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

2

+ T + 3 27 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

2

+ 2T + 3 21 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

2

+ 3T + 1 29 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

2

+ 4T + 1 33 ≡ 1 (mod 2)

(10)

q δ Q(T ) g h

K

5 5

T

5

+ 3T

4

+ T

3

+ 2T + 2

2

51 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 3T + 4 27 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 4T + 2 21 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ T

2

+ T + 2 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ T

2

+ 2T + 1 49 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ T

2

+ 2T + 3 35 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ T

2

+ 3T + 3 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ T

2

+ 4T + 4 39 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 2T

2

+ 1 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 2T

2

+ 3T + 1 51 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 2T

2

+ 4T + 3 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 3T

2

+ 3 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 3T

2

+ 4 27 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 3T

2

+ 2T + 2 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 3T

2

+ 3T + 2 13 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 3T

2

+ 3T + 3 41 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 3T

2

+ 4T + 1 39 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 4T

2

+ 2 41 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 4T

2

+ 3 9 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 4T

2

+ 3T + 4 53 ≡ 1 (mod 2)

T

5

+ 3T

4

+ T

3

+ 4T

2

+ 4T + 1 21 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 1 41 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 2 9 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 3T + 2 53 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 4T + 2 21 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ T

2

+ 2 21 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ T

2

+ T + 1 29 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ T

2

+ 2T + 2 33 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ T

2

+ 2T + 3 11 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ T

2

+ 2T + 4 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ T

2

+ 3T + 1 51 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ T

2

+ 4T + 1 27 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 2T

2

+ 4 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 2T

2

+ T + 3 39 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 2T

2

+ 3T + 2 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 2T

2

+ 4T + 1 35 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 2T

2

+ 4T + 4 49 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 3T

2

+ T + 1 51 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 3T

2

+ 2T + 1 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 3T

2

+ 3T + 2 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 4T

2

+ T + 3 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 4T

2

+ 2T + 1 13 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 4T

2

+ 2T + 2 41 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 4T

2

+ 3T + 3 39 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 4T

2

+ 4T + 3 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 2T

3

+ 4T

2

+ 4T + 4 27 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ T + 1 53 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 2T + 4 21 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 3T + 1 41 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 3T + 2 9 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ T

2

+ 3 29 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ T

2

+ T + 2 33 ≡ 1 (mod 2)

(11)

q δ Q(T ) g h

K

5 5

T

5

+ 3T

4

+ 3T

3

+ T

2

+ T + 3

2

11 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ T

2

+ T + 4 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ T

2

+ 2T + 4 51 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ T

2

+ 3T + 2 27 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ T

2

+ 4T + 1 21 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 2T

2

+ 2 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 2T

2

+ T + 4 39 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 2T

2

+ 3T + 4 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 2T

2

+ 4T + 1 35 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 2T

2

+ 4T + 4 49 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 3T

2

+ 2T + 1 51 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 3T

2

+ 3T + 4 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 3T

2

+ 4T + 3 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 4T

2

+ 3 39 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 4T

2

+ T + 1 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 4T

2

+ T + 2 27 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 4T

2

+ 3T + 2 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 4T

2

+ 4T + 3 13 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 3T

3

+ 4T

2

+ 4T + 4 41 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 4 21 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ T + 4 29 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 2T + 1 33 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 2T + 2 11 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 2T + 3 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 3T + 1 51 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 4T + 2 27 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ T

2

+ 3 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ T

2

+ T + 1 49 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ T

2

+ T + 3 35 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ T

2

+ 2T + 2 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ T

2

+ 3T + 2 39 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 2T

2

+ 1 51 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 2T

2

+ T + 2 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 2T

2

+ 2T + 4 23 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 3T

2

+ 2 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 3T

2

+ 3 27 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 3T

2

+ 2T + 4 19 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 3T

2

+ 3T + 3 13 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 3T

2

+ 3T + 4 41 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 3T

2

+ 4T + 1 39 ≡ 1 (mod 2)

T

5

+ 3T

4

+ 4T

3

+ 4T

2

+ T + 1 53 ≡ 1 (mod 2)

Gambar

Table 2. Divisor class numbers of Kummer extensions with ` = 3
Table 2. Divisor class numbers of Kummer extensions with ` = 3 (Cont’d)
Table 3. Divisor class numbers of Kummer extensions with ` = 5
Table 4. Divisor class numbers of Kummer extensions with ` = 7
+7

Referensi

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