# 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 = 14 = 2*7 (relative mode)

Note, that for n ≡ 2 mod 4 the relative cyclotomic units are trivial!

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

The rank (number of elements in basis) is 0

## Cyclotomic Units Base Representations for n = 14 = 2*7 (relative mode)

Note, that for n ≡ 2 mod 4 the relative cyclotomic units are trivial!
 v(14,1) v(14,3) v(14,5) v(14,9) v(14,11) v(14,13)

## Cyclotomic Units Base Representations for n = 14 = 2*7 (relative mode)

See Algorithm 2.4 for details

Note, that for n ≡ 2 mod 4 the relative cyclotomic units are trivial!

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(14,1) = 1
3: v(14,3) = 1
5: v(14,5) = 1
9: v(14,9) = 1
11: v(14,11) = 1
13: v(14,13) = 1
The rank is: 0

## Cyclotomic Units - Methods for First Development for n = 14 = 2*7

Note, that for n ≡ 2 mod 4 the relative cyclotomic units are trivial!

Compare with Algorithm 1.2 and 2.2

1: v(14,1) - method Z-2 (Case IV - A12)
3: v(14,3) - method Z-2 (Case IV - A12)
5: v(14,5) - method Z-2 (Case IV - A12)
9: v(14,9) - method Z-2 (Case IV - A12)
11: v(14,11) - method Z-2 (Case IV - A12)
13: v(14,13) - method Z-2 (Case IV - A12)