# 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 = 9 = 3^2 (relative mode)

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

v(9,4), v(9,7)

The rank (number of elements in basis) is 2

## Cyclotomic Units Base Representations for n = 9 = 3^2 (relative mode)

 v(9,4) v(9,7) v(9,1) v(9,2) 1 v(9,5) 1 v(9,8)

## Cyclotomic Units Base Representations for n = 9 = 3^2 (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(9,1) = 1
2: v(9,2) = v(9,7)
4: v(9,4) is in basis
5: v(9,5) = v(9,4)
7: v(9,7) is in basis
8: v(9,8) = 1
The rank is: 2

## Cyclotomic Units - Methods for First Development for n = 9 = 3^2

Compare with Algorithm 2.2

1: v(9,1) - method T (Case IV,iii - A22)
2: v(9,2) - method S (Case IV,i - A22)
4: v(9,4) - method B (Case IV,ii - A22)
5: v(9,5) - method S (Case IV,i - A22)
7: v(9,7) - method B (Case IV,ii - A22)
8: v(9,8) - method S (Case IV,i - A22)