# 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 = 13 (prime) (relative mode)

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

v(13,2), v(13,3), v(13,4), v(13,5), v(13,6)

The rank (number of elements in basis) is 5

## Cyclotomic Units Base Representations for n = 13 (prime) (relative mode)

 v(13,2) v(13,3) v(13,4) v(13,5) v(13,6) v(13,1) v(13,7) 1 v(13,8) 1 v(13,9) 1 v(13,10) 1 v(13,11) 1 v(13,12)

## Cyclotomic Units Base Representations for n = 13 (prime) (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(13,1) = 1
2: v(13,2) is in basis
3: v(13,3) is in basis
4: v(13,4) is in basis
5: v(13,5) is in basis
6: v(13,6) is in basis
7: v(13,7) = v(13,6)
8: v(13,8) = v(13,5)
9: v(13,9) = v(13,4)
10: v(13,10) = v(13,3)
11: v(13,11) = v(13,2)
12: v(13,12) = 1
The rank is: 5

## Cyclotomic Units - Methods for First Development for n = 13 (prime)

Compare with Algorithm 2.2

1: v(13,1) - method T (Case III, A22)
2: v(13,2) - method B (Case III, A22)
3: v(13,3) - method B (Case III, A22)
4: v(13,4) - method B (Case III, A22)
5: v(13,5) - method B (Case III, A22)
6: v(13,6) - method B (Case III, A22)
7: v(13,7) - method S (Case III, A22)
8: v(13,8) - method S (Case III, A22)
9: v(13,9) - method S (Case III, A22)
10: v(13,10) - method S (Case III, A22)
11: v(13,11) - method S (Case III, A22)
12: v(13,12) - method S (Case III, A22)