# 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 = 16 = 2^4

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(16,9), v(8,5), v(16,11)

The rank (number of elements in basis) is 3

## Cyclotomic Units Base Representations for n = 16 = 2^4

 v(16,9) v(8,5) v(16,11) v(16,1) v(8,1) v(16,3) 1 1 -1 v(4,1) v(16,5) 1 v(8,3) 1 v(16,7) 1 v(2,1) v(4,3) v(16,13) 1 1 -1 v(8,7) v(16,15)

## Cyclotomic Units Base Representations for n = 16 = 2^4

See Algorithm 2.4 for details

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

1: v(16,1) = 1
2: v(8,1) = 1
3: v(16,3) = v(16,9) v(16,11)^-1 v(8,5)
4: v(4,1) = 1
5: v(16,5) = v(16,11)
6: v(8,3) = v(8,5)
7: v(16,7) = v(16,9)
8: v(2,1) = 1
9: v(16,9) is in basis
10: v(8,5) is in basis
11: v(16,11) is in basis
12: v(4,3) = 1
13: v(16,13) = v(16,9) v(16,11)^-1 v(8,5)
14: v(8,7) = 1
15: v(16,15) = 1
The rank is: 3

## Cyclotomic Units - Methods for First Development for n = 16 = 2^4

Compare with Algorithm 2.2

1: v(16,1) - method T (Case IV,iii - A22)
2: v(8,1) - method T (Case IV,iii - A22)
3: v(16,3) - method Y-2 (Case IV,iv - A22)
4: v(4,1) - method T (Case II - A22)
5: v(16,5) - method S (Case IV,i - A22)
6: v(8,3) - method S (Case IV,i - A22)
7: v(16,7) - method S (Case IV,i - A22)
8: v(2,1) - method T (Case II - A22)
9: v(16,9) - method B (Case IV,ii - A22)
10: v(8,5) - method B (Case IV,ii - A22)
11: v(16,11) - method B (Case IV,ii - A22)
12: v(4,3) - method T (Case II - A22)
13: v(16,13) - method S (Case IV,i - A22)
14: v(8,7) - method S (Case IV,i - A22)
15: v(16,15) - method S (Case IV,i - A22)