# 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 = 12 = 2^2*3

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(12,1)

The rank (number of elements in basis) is 1

## Cyclotomic Units Base Representations for n = 12 = 2^2*3

 v(12,1) v(6,1) v(4,1) v(3,1) v(12,5) -1 v(2,1) v(12,7) -1 v(3,2) v(4,3) v(6,5) v(12,11) 1

## Cyclotomic Units Base Representations for n = 12 = 2^2*3

See Algorithm 2.4 for details

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

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

## Cyclotomic Units - Methods for First Development for n = 12 = 2^2*3

Compare with Algorithm 1.2 and 2.2

1: v(12,1) - method B (Case V,i - A12)
2: v(6,1) - method Z-2 (Case IV - A12)
3: v(4,1) - method T (Case II - A22)
4: v(3,1) - method T (Case III, A22)
5: v(12,5) - method Z-3 (Case V,iv - A12)
6: v(2,1) - method T (Case II - A22)
7: v(12,7) - method Z-2 (Case V,iv - A12)
8: v(3,2) - method S (Case III, A22)
9: v(4,3) - method T (Case II - A22)
10: v(6,5) - method Z-2 (Case IV - A12)
11: v(12,11) - method Z-2 (Case V,iv - A12)