# 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

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(7,2), v(7,3)

The rank (number of elements in basis) is 2

## Cyclotomic Units Base Representations for n = 14 = 2*7

 v(7,2) v(7,3) v(14,1) -1 v(7,1) v(14,3) -1 1 v(14,5) 1 v(2,1) v(7,4) 1 v(14,9) 1 v(7,5) 1 v(14,11) -1 1 v(7,6) v(14,13) -1

## Cyclotomic Units Base Representations for n = 14 = 2*7

See Algorithm 2.4 for details

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

1: v(14,1) = v(7,3)^-1
2: v(7,1) = 1
3: v(14,3) = v(7,2)^-1 v(7,3)
4: v(7,2) is in basis
5: v(14,5) = v(7,2)
6: v(7,3) is in basis
7: v(2,1) = 1
8: v(7,4) = v(7,3)
9: v(14,9) = v(7,2)
10: v(7,5) = v(7,2)
11: v(14,11) = v(7,2)^-1 v(7,3)
12: v(7,6) = 1
13: v(14,13) = v(7,3)^-1
The rank is: 2

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

Compare with Algorithm 1.2 and 2.2

1: v(14,1) - method Z-2 (Case IV - A12)
2: v(7,1) - method T (Case III, A22)
3: v(14,3) - method Z-2 (Case IV - A12)
4: v(7,2) - method B (Case III, A22)
5: v(14,5) - method Z-2 (Case IV - A12)
6: v(7,3) - method B (Case III, A22)
7: v(2,1) - method T (Case II - A22)
8: v(7,4) - method S (Case III, A22)
9: v(14,9) - method Z-2 (Case IV - A12)
10: v(7,5) - method S (Case III, A22)
11: v(14,11) - method Z-2 (Case IV - A12)
12: v(7,6) - method S (Case III, A22)
13: v(14,13) - method Z-2 (Case IV - A12)