# 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 (relative mode)

Note that the union of all bases over all n extends to a universal basis

v(12,1)

The rank (number of elements in basis) is 1

## Cyclotomic Units Base Representations for n = 12 = 2^2*3 (relative mode)

 v(12,1) v(12,5) -1 v(12,7) -1 v(12,11) 1

## Cyclotomic Units Base Representations for n = 12 = 2^2*3 (relative mode)

See Algorithm 2.4 for details

Note that equality is modulo multiplication by an nth unit root and modulo dth cyclotomic units where d is a proper divisor of n.

1: v(12,1) is in basis
5: v(12,5) = v(12,1)^-1
7: v(12,7) = v(12,1)^-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)
5: v(12,5) - method Z-3 (Case V,iv - A12)
7: v(12,7) - method Z-2 (Case V,iv - A12)
11: v(12,11) - method Z-2 (Case V,iv - A12)