# 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 = 15 = 3*5

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(15,1), v(5,2), v(15,13)

The rank (number of elements in basis) is 3

## Cyclotomic Units Base Representations for n = 15 = 3*5

 v(15,1) v(5,2) v(15,13) v(15,2) 1 v(5,1) v(15,4) -1 -1 v(3,1) v(15,7) 1 -1 v(15,8) 1 -1 v(5,3) 1 v(3,2) v(15,11) -1 -1 v(5,4) v(15,14) 1

## Cyclotomic Units Base Representations for n = 15 = 3*5

See Algorithm 2.4 for details

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

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

## Cyclotomic Units - Methods for First Development for n = 15 = 3*5

Compare with Algorithm 1.2 and 2.2

1: v(15,1) - method B (Case V,i - A12)
2: v(15,2) - method Z-3 (Case V,iv - A12)
3: v(5,1) - method T (Case III, A22)
4: v(15,4) - method Z-5 (Case V,iv - A12)
5: v(3,1) - method T (Case III, A22)
6: v(5,2) - method B (Case III, A22)
7: v(15,7) - method S (Case V,ii - A12 for p=5)
8: v(15,8) - method Z-3 (Case V,iv - A12)
9: v(5,3) - method S (Case III, A22)
10: v(3,2) - method S (Case III, A22)
11: v(15,11) - method Z-3 (Case V,iv - A12)
12: v(5,4) - method S (Case III, A22)
13: v(15,13) - method B (Case V,iii - A12)
14: v(15,14) - method Z-3 (Case V,iv - A12)