# 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) © 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).