# 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 = 16 = 2^4

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(16,9), v(8,5), v(16,11)

The rank (number of elements in basis) is 3

## Cyclotomic Units Base Representations for n = 16 = 2^4

 v(16,9) v(8,5) v(16,11) v(16,1) v(8,1) v(16,3) 1 1 -1 v(4,1) v(16,5) 1 v(8,3) 1 v(16,7) 1 v(2,1) v(4,3) v(16,13) 1 1 -1 v(8,7) v(16,15)

## Cyclotomic Units Base Representations for n = 16 = 2^4

See Algorithm 2.4 for details

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

1: v(16,1) = 1
2: v(8,1) = 1
3: v(16,3) = v(16,9) v(16,11)^-1 v(8,5)
4: v(4,1) = 1
5: v(16,5) = v(16,11)
6: v(8,3) = v(8,5)
7: v(16,7) = v(16,9)
8: v(2,1) = 1
9: v(16,9) is in basis
10: v(8,5) is in basis
11: v(16,11) is in basis
12: v(4,3) = 1
13: v(16,13) = v(16,9) v(16,11)^-1 v(8,5)
14: v(8,7) = 1
15: v(16,15) = 1
The rank is: 3

## Cyclotomic Units - Methods for First Development for n = 16 = 2^4

Compare with Algorithm 2.2

1: v(16,1) - method T (Case IV,iii - A22)
2: v(8,1) - method T (Case IV,iii - A22)
3: v(16,3) - method Y-2 (Case IV,iv - A22)
4: v(4,1) - method T (Case II - A22)
5: v(16,5) - method S (Case IV,i - A22)
6: v(8,3) - method S (Case IV,i - A22)
7: v(16,7) - method S (Case IV,i - A22)
8: v(2,1) - method T (Case II - A22)
9: v(16,9) - method B (Case IV,ii - A22)
10: v(8,5) - method B (Case IV,ii - A22)
11: v(16,11) - method B (Case IV,ii - A22)
12: v(4,3) - method T (Case II - A22)
13: v(16,13) - method S (Case IV,i - A22)
14: v(8,7) - method S (Case IV,i - A22)
15: v(16,15) - method S (Case IV,i - A22) © by: Marc Conrad, 2016. If you want me to come to your organisation and talk about cyclotomic units please contact me. The material on this page is presented "as is". There is no warranty implied by presenting this stuff. Feel free to use and modify the material for your own teaching. When doing so please link to this web site (http://perisic.com/cyclotomic). In acadmic publications cite this page as: Conrad, Marc (2016) Cyclotomic Units - Relations and Computations (online), available at: http://perisic.com/cyclotomic. The webspace for this project is kindly provided by the Perisic Guesthouse (www.perisic.com).