# Cyclotomic Unit Calculator

## Definitions

• εn := ei/n is an nth root of unity.
• For n=pα a prime power: v(n,a) := (1 - εna)/ (1 - εn)
• For all other n: v(n,a) := 1 - εna
Cyclotomic Unit Calculation

## Basis for Cyclotomic Units for n = 105 = 3*5*7

Note that this basis extends to a universal basis, i.e the basis for d is a subset of the basis for n iff d|n.

v(35,1), v(21,1), v(15,1), v(35,3), v(21,4), v(35,8), v(7,2), v(35,11), v(5,2), v(105,43), v(7,3), v(105,46), v(35,18), v(105,58), v(105,61), v(35,23), v(105,73), v(35,26), v(105,88), v(15,13), v(21,19), v(35,33), v(105,103)

The rank (number of elements in basis) is 23

## Cyclotomic Units Base Representations for n = 105 = 3*5*7

 v(35,1) v(21,1) v(15,1) v(35,3) v(21,4) v(35,8) v(7,2) v(35,11) v(5,2) v(105,43) v(7,3) v(105,46) v(35,18) v(105,58) v(105,61) v(35,23) v(105,73) v(35,26) v(105,88) v(15,13) v(21,19) v(35,33) v(105,103) v(105,1) 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 v(105,2) 1 v(105,4) 1 -1 1 -1 1 -1 1 -1 1 v(35,2) 1 v(105,8) 1 -1 -1 v(21,2) 1 v(105,11) -1 1 -1 v(35,4) -1 -1 -1 1 -1 v(105,13) 1 1 -1 -1 -1 -1 1 -1 v(15,2) 1 v(7,1) v(105,16) -1 1 1 1 -1 -1 1 -1 -1 -1 1 v(105,17) 1 v(35,6) -1 1 -1 -1 1 1 1 v(105,19) 1 -1 -1 1 -1 1 -1 -1 1 -1 v(5,1) v(105,22) -1 -1 -1 -1 1 -1 1 1 1 1 -1 1 v(105,23) 1 1 1 -1 1 -1 1 v(21,5) 1 -1 -1 v(105,26) -1 -1 1 v(35,9) 1 v(15,4) -1 -1 v(105,29) -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 v(105,31) -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 v(105,32) 1 v(105,34) -1 -1 1 -1 1 -1 1 1 1 -1 1 v(3,1) v(35,12) 1 v(105,37) -1 1 -1 v(105,38) -1 1 -1 v(35,13) -1 -1 1 -1 -1 -1 v(21,8) -1 -1 v(105,41) -1 2 -1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 v(105,44) 1 v(105,47) 1 v(35,16) 1 -1 -1 -1 v(15,7) 1 -1 v(21,10) -1 1 v(35,17) 1 v(105,52) 1 -1 1 1 -1 -1 -1 v(105,53) 1 -1 1 1 -1 -1 -1 v(21,11) -1 1 v(15,8) 1 -1 v(35,19) 1 -1 -1 -1 v(105,59) 1 v(7,4) 1 v(105,62) 1 v(5,3) 1 v(105,64) -1 2 -1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 v(21,13) -1 -1 v(35,22) -1 -1 1 -1 -1 -1 v(105,67) -1 1 -1 v(105,68) -1 1 -1 v(3,2) v(105,71) -1 -1 1 -1 1 -1 1 1 1 -1 1 v(35,24) 1 v(105,74) -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 v(7,5) 1 v(105,76) -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 v(15,11) -1 -1 v(105,79) -1 -1 1 v(21,16) 1 -1 -1 v(35,27) 1 v(105,82) 1 1 1 -1 1 -1 1 v(105,83) -1 -1 -1 -1 1 -1 1 1 1 1 -1 1 v(5,4) v(21,17) 1 v(105,86) 1 -1 -1 1 -1 1 -1 -1 1 -1 v(35,29) -1 1 -1 -1 1 1 1 v(105,89) -1 1 1 1 -1 -1 1 -1 -1 -1 1 v(7,6) v(105,92) 1 1 -1 -1 -1 -1 1 -1 v(35,31) -1 -1 -1 1 -1 v(105,94) -1 1 -1 v(35,32) 1 v(105,97) 1 -1 -1 v(15,14) 1 v(21,20) 1 v(105,101) 1 -1 1 -1 1 -1 1 -1 1 v(35,34) 1 v(105,104) 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1

## Cyclotomic Units Base Representations for n = 105 = 3*5*7

See Algorithm 2.4 for details

Note that equality is modulo multiplication by an nth unit root.

1: v(105,1) = v(105,43)^-1 v(105,46)^-1 v(105,61)^-1 v(105,88)^-1 v(105,103)^-1 v(35,1) v(35,3)^-1 v(35,23)^-1 v(21,1) v(21,19) v(15,1) v(7,3) v(5,2)
2: v(105,2) = v(105,103)
3: v(35,1) is in basis
4: v(105,4) = v(105,46)^-1 v(105,73) v(105,88)^-1 v(35,1) v(35,3)^-1 v(21,4) v(21,19) v(7,2)^-1 v(7,3)
5: v(21,1) is in basis
6: v(35,2) = v(35,33)
7: v(15,1) is in basis
8: v(105,8) = v(105,43)^-1 v(35,8) v(35,26)^-1
9: v(35,3) is in basis
10: v(21,2) = v(21,19)
11: v(105,11) = v(105,46)^-1 v(35,8)^-1 v(35,11)
12: v(35,4) = v(35,3)^-1 v(35,11)^-1 v(35,18)^-1 v(7,2)^-1 v(7,3)
13: v(105,13) = v(105,43)^-1 v(105,58)^-1 v(105,73)^-1 v(105,88)^-1 v(105,103)^-1 v(15,1) v(15,13) v(5,2)
14: v(15,2) = v(15,13)
15: v(7,1) = 1
16: v(105,16) = v(105,58)^-1 v(105,61) v(105,103) v(35,3) v(35,11) v(35,26)^-1 v(35,33)^-1 v(21,1)^-1 v(21,19)^-1 v(7,2) v(7,3)^-1
17: v(105,17) = v(105,88)
18: v(35,6) = v(35,1)^-1 v(35,3) v(35,18) v(35,23) v(35,33) v(7,3)^-1 v(5,2)^-1
19: v(105,19) = v(105,58) v(105,61)^-1 v(105,103)^-1 v(35,3)^-1 v(35,11)^-1 v(35,18)^-1 v(35,23)^-1 v(21,1) v(21,19) v(7,3)
20: v(21,4) is in basis
21: v(5,1) = 1
22: v(105,22) = v(105,43) v(105,58) v(105,73) v(105,88) v(105,103) v(35,3)^-1 v(35,8)^-1 v(35,18)^-1 v(35,26) v(15,1)^-1 v(15,13)^-1 v(7,2)^-1
23: v(105,23) = v(105,58)^-1 v(35,3) v(35,11) v(35,18) v(35,23) v(7,2) v(7,3)^-1
24: v(35,8) is in basis
25: v(21,5) = v(21,19)^-1 v(7,2) v(7,3)^-1
26: v(105,26) = v(105,61)^-1 v(35,3)^-1 v(35,26)
27: v(35,9) = v(35,26)
28: v(15,4) = v(15,1)^-1 v(5,2)^-1
29: v(105,29) = v(105,43) v(105,46)^-1 v(105,58) v(105,61)^-1 v(105,73) v(35,3)^-1 v(35,8)^-1 v(35,26) v(21,4) v(21,19) v(15,13)^-1 v(7,2)^-1
30: v(7,2) is in basis
31: v(105,31) = v(105,46) v(105,73)^-1 v(105,88) v(35,1)^-1 v(35,3) v(35,8) v(35,11)^-1 v(35,23) v(35,33) v(21,4)^-1 v(21,19)^-1 v(5,2)^-1
32: v(105,32) = v(105,73)
33: v(35,11) is in basis
34: v(105,34) = v(105,43) v(105,46) v(105,61) v(105,88) v(105,103) v(35,3) v(21,1)^-1 v(21,19)^-1 v(15,1)^-1 v(7,3)^-1 v(5,2)^-1
35: v(3,1) = 1
36: v(35,12) = v(35,23)
37: v(105,37) = v(105,103)^-1 v(35,11)^-1 v(35,33)
38: v(105,38) = v(105,73)^-1 v(35,1)^-1 v(35,3)
39: v(35,13) = v(35,3)^-1 v(35,8)^-1 v(35,18)^-1 v(35,23)^-1 v(35,33)^-1 v(5,2)
40: v(21,8) = v(21,1)^-1 v(7,2)^-1
41: v(105,41) = v(105,43)^-1 v(105,46) v(105,58)^-1 v(105,61) v(105,73)^-1 v(35,1)^-1 v(35,3)^2 v(35,8) v(35,18) v(35,23) v(35,26)^-1 v(21,4)^-1 v(21,19)^-1 v(15,13) v(7,2) v(7,3)^-1 v(5,2)^-1
42: v(5,2) is in basis
43: v(105,43) is in basis
44: v(105,44) = v(105,61)
45: v(7,3) is in basis
46: v(105,46) is in basis
47: v(105,47) = v(105,58)
48: v(35,16) = v(35,23)^-1 v(35,26)^-1 v(35,33)^-1 v(7,2)
49: v(15,7) = v(15,13)^-1 v(5,2)
50: v(21,10) = v(21,4)^-1 v(7,3)
51: v(35,17) = v(35,18)
52: v(105,52) = v(105,88)^-1 v(35,1) v(35,3)^-1 v(35,23)^-1 v(35,33)^-1 v(7,3) v(5,2)
53: v(105,53) = v(105,88)^-1 v(35,1) v(35,3)^-1 v(35,23)^-1 v(35,33)^-1 v(7,3) v(5,2)
54: v(35,18) is in basis
55: v(21,11) = v(21,4)^-1 v(7,3)
56: v(15,8) = v(15,13)^-1 v(5,2)
57: v(35,19) = v(35,23)^-1 v(35,26)^-1 v(35,33)^-1 v(7,2)
58: v(105,58) is in basis
59: v(105,59) = v(105,46)
60: v(7,4) = v(7,3)
61: v(105,61) is in basis
62: v(105,62) = v(105,43)
63: v(5,3) = v(5,2)
64: v(105,64) = v(105,43)^-1 v(105,46) v(105,58)^-1 v(105,61) v(105,73)^-1 v(35,1)^-1 v(35,3)^2 v(35,8) v(35,18) v(35,23) v(35,26)^-1 v(21,4)^-1 v(21,19)^-1 v(15,13) v(7,2) v(7,3)^-1 v(5,2)^-1
65: v(21,13) = v(21,1)^-1 v(7,2)^-1
66: v(35,22) = v(35,3)^-1 v(35,8)^-1 v(35,18)^-1 v(35,23)^-1 v(35,33)^-1 v(5,2)
67: v(105,67) = v(105,73)^-1 v(35,1)^-1 v(35,3)
68: v(105,68) = v(105,103)^-1 v(35,11)^-1 v(35,33)
69: v(35,23) is in basis
70: v(3,2) = 1
71: v(105,71) = v(105,43) v(105,46) v(105,61) v(105,88) v(105,103) v(35,3) v(21,1)^-1 v(21,19)^-1 v(15,1)^-1 v(7,3)^-1 v(5,2)^-1
72: v(35,24) = v(35,11)
73: v(105,73) is in basis
74: v(105,74) = v(105,46) v(105,73)^-1 v(105,88) v(35,1)^-1 v(35,3) v(35,8) v(35,11)^-1 v(35,23) v(35,33) v(21,4)^-1 v(21,19)^-1 v(5,2)^-1
75: v(7,5) = v(7,2)
76: v(105,76) = v(105,43) v(105,46)^-1 v(105,58) v(105,61)^-1 v(105,73) v(35,3)^-1 v(35,8)^-1 v(35,26) v(21,4) v(21,19) v(15,13)^-1 v(7,2)^-1
77: v(15,11) = v(15,1)^-1 v(5,2)^-1
78: v(35,26) is in basis
79: v(105,79) = v(105,61)^-1 v(35,3)^-1 v(35,26)
80: v(21,16) = v(21,19)^-1 v(7,2) v(7,3)^-1
81: v(35,27) = v(35,8)
82: v(105,82) = v(105,58)^-1 v(35,3) v(35,11) v(35,18) v(35,23) v(7,2) v(7,3)^-1
83: v(105,83) = v(105,43) v(105,58) v(105,73) v(105,88) v(105,103) v(35,3)^-1 v(35,8)^-1 v(35,18)^-1 v(35,26) v(15,1)^-1 v(15,13)^-1 v(7,2)^-1
84: v(5,4) = 1
85: v(21,17) = v(21,4)
86: v(105,86) = v(105,58) v(105,61)^-1 v(105,103)^-1 v(35,3)^-1 v(35,11)^-1 v(35,18)^-1 v(35,23)^-1 v(21,1) v(21,19) v(7,3)
87: v(35,29) = v(35,1)^-1 v(35,3) v(35,18) v(35,23) v(35,33) v(7,3)^-1 v(5,2)^-1
88: v(105,88) is in basis
89: v(105,89) = v(105,58)^-1 v(105,61) v(105,103) v(35,3) v(35,11) v(35,26)^-1 v(35,33)^-1 v(21,1)^-1 v(21,19)^-1 v(7,2) v(7,3)^-1
90: v(7,6) = 1
91: v(15,13) is in basis
92: v(105,92) = v(105,43)^-1 v(105,58)^-1 v(105,73)^-1 v(105,88)^-1 v(105,103)^-1 v(15,1) v(15,13) v(5,2)
93: v(35,31) = v(35,3)^-1 v(35,11)^-1 v(35,18)^-1 v(7,2)^-1 v(7,3)
94: v(105,94) = v(105,46)^-1 v(35,8)^-1 v(35,11)
95: v(21,19) is in basis
96: v(35,32) = v(35,3)
97: v(105,97) = v(105,43)^-1 v(35,8) v(35,26)^-1
98: v(15,14) = v(15,1)
99: v(35,33) is in basis
100: v(21,20) = v(21,1)
101: v(105,101) = v(105,46)^-1 v(105,73) v(105,88)^-1 v(35,1) v(35,3)^-1 v(21,4) v(21,19) v(7,2)^-1 v(7,3)
102: v(35,34) = v(35,1)
103: v(105,103) is in basis
104: v(105,104) = v(105,43)^-1 v(105,46)^-1 v(105,61)^-1 v(105,88)^-1 v(105,103)^-1 v(35,1) v(35,3)^-1 v(35,23)^-1 v(21,1) v(21,19) v(15,1) v(7,3) v(5,2)
The rank is: 23

## Cyclotomic Units - Methods for First Development for n = 105 = 3*5*7

Compare with Algorithm 1.2 and 2.2

1: v(105,1) - method E (Case V,i - A12)
2: v(105,2) - method Z-3 (Case V,iv - A12)
3: v(35,1) - method B (Case V,i - A12)
4: v(105,4) - method Z-5 (Case V,iv - A12)
5: v(21,1) - method B (Case V,i - A12)
6: v(35,2) - method S (Case V,ii - A12 for p=5)
7: v(15,1) - method B (Case V,i - A12)
8: v(105,8) - method Z-3 (Case V,iv - A12)
9: v(35,3) - method B (Case V,iii - A12)
10: v(21,2) - method Z-3 (Case V,iv - A12)
11: v(105,11) - method Z-3 (Case V,iv - A12)
12: v(35,4) - method Z-5 (Case V,iv - A12)
13: v(105,13) - method Z-7 (Case V,iv - A12)
14: v(15,2) - method Z-3 (Case V,iv - A12)
15: v(7,1) - method T (Case III, A22)
16: v(105,16) - method S (Case V,ii - A12 for p=7)
17: v(105,17) - method Z-3 (Case V,iv - A12)
18: v(35,6) - method Z-7 (Case V,iv - A12)
19: v(105,19) - method Z-5 (Case V,iv - A12)
20: v(21,4) - method B (Case V,iii - A12)
21: v(5,1) - method T (Case III, A22)
22: v(105,22) - method S (Case V,ii - A12 for p=5)
23: v(105,23) - method Z-3 (Case V,iv - A12)
24: v(35,8) - method B (Case V,iii - A12)
25: v(21,5) - method Z-3 (Case V,iv - A12)
26: v(105,26) - method Z-3 (Case V,iv - A12)
27: v(35,9) - method Z-5 (Case V,iv - A12)
28: v(15,4) - method Z-5 (Case V,iv - A12)
29: v(105,29) - method Z-3 (Case V,iv - A12)
30: v(7,2) - method B (Case III, A22)
31: v(105,31) - method S (Case V,ii - A12 for p=7)
32: v(105,32) - method Z-3 (Case V,iv - A12)
33: v(35,11) - method B (Case V,iii - A12)
34: v(105,34) - method Z-5 (Case V,iv - A12)
35: v(3,1) - method T (Case III, A22)
36: v(35,12) - method S (Case V,ii - A12 for p=5)
37: v(105,37) - method S (Case V,ii - A12 for p=5)
38: v(105,38) - method Z-3 (Case V,iv - A12)
39: v(35,13) - method Z-7 (Case V,iv - A12)
40: v(21,8) - method Z-3 (Case V,iv - A12)
41: v(105,41) - method Z-3 (Case V,iv - A12)
42: v(5,2) - method B (Case III, A22)
43: v(105,43) - method B (Case V,iii - A12)
44: v(105,44) - method Z-3 (Case V,iv - A12)
45: v(7,3) - method B (Case III, A22)
46: v(105,46) - method B (Case V,iii - A12)
47: v(105,47) - method Z-3 (Case V,iv - A12)
48: v(35,16) - method S (Case V,ii - A12 for p=7)
49: v(15,7) - method S (Case V,ii - A12 for p=5)
50: v(21,10) - method S (Case V,ii - A12 for p=7)
51: v(35,17) - method S (Case V,ii - A12 for p=5)
52: v(105,52) - method S (Case V,ii - A12 for p=5)
53: v(105,53) - method Z-3 (Case V,iv - A12)
54: v(35,18) - method B (Case V,iii - A12)
55: v(21,11) - method Z-3 (Case V,iv - A12)
56: v(15,8) - method Z-3 (Case V,iv - A12)
57: v(35,19) - method Z-5 (Case V,iv - A12)
58: v(105,58) - method B (Case V,iii - A12)
59: v(105,59) - method Z-3 (Case V,iv - A12)
60: v(7,4) - method S (Case III, A22)
61: v(105,61) - method B (Case V,iii - A12)
62: v(105,62) - method Z-3 (Case V,iv - A12)
63: v(5,3) - method S (Case III, A22)
64: v(105,64) - method Z-5 (Case V,iv - A12)
65: v(21,13) - method Z-7 (Case V,iv - A12)
66: v(35,22) - method S (Case V,ii - A12 for p=5)
67: v(105,67) - method S (Case V,ii - A12 for p=5)
68: v(105,68) - method Z-3 (Case V,iv - A12)
69: v(35,23) - method B (Case V,iii - A12)
70: v(3,2) - method S (Case III, A22)
71: v(105,71) - method Z-3 (Case V,iv - A12)
72: v(35,24) - method Z-5 (Case V,iv - A12)
73: v(105,73) - method B (Case V,iii - A12)
74: v(105,74) - method Z-3 (Case V,iv - A12)
75: v(7,5) - method S (Case III, A22)
76: v(105,76) - method Z-7 (Case V,iv - A12)
77: v(15,11) - method Z-3 (Case V,iv - A12)
78: v(35,26) - method B (Case V,iii - A12)
79: v(105,79) - method Z-5 (Case V,iv - A12)
80: v(21,16) - method S (Case V,ii - A12 for p=7)
81: v(35,27) - method S (Case V,ii - A12 for p=5)
82: v(105,82) - method S (Case V,ii - A12 for p=5)
83: v(105,83) - method Z-3 (Case V,iv - A12)
84: v(5,4) - method S (Case III, A22)
85: v(21,17) - method Z-3 (Case V,iv - A12)
86: v(105,86) - method Z-3 (Case V,iv - A12)
87: v(35,29) - method Z-5 (Case V,iv - A12)
88: v(105,88) - method B (Case V,iii - A12)
89: v(105,89) - method Z-3 (Case V,iv - A12)
90: v(7,6) - method S (Case III, A22)
91: v(15,13) - method B (Case V,iii - A12)
92: v(105,92) - method Z-3 (Case V,iv - A12)
93: v(35,31) - method S (Case V,ii - A12 for p=7)
94: v(105,94) - method Z-5 (Case V,iv - A12)
95: v(21,19) - method B (Case V,iii - A12)
96: v(35,32) - method S (Case V,ii - A12 for p=5)
97: v(105,97) - method S (Case V,ii - A12 for p=5)
98: v(15,14) - method Z-3 (Case V,iv - A12)
99: v(35,33) - method B (Case V,iii - A12)
100: v(21,20) - method Z-3 (Case V,iv - A12)
101: v(105,101) - method Z-3 (Case V,iv - A12)
102: v(35,34) - method Z-5 (Case V,iv - A12)
103: v(105,103) - method B (Case V,iii - A12)
104: v(105,104) - method Z-3 (Case V,iv - A12)