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ID: INPUT{id.yaml}

Title: >
  Teichmüller representatives in $\mathbb{Z}_p$

Definition: >
  Let $p$ be a rational prime,
  let $q=p$ for $p>2$ and $q=4$ for $p=2$,
  let $G = (\mathbb{Z}/q\mathbb{Z})^\times$, 
  and let $\omega: G \to \mathbb{Z}_p^*$ be
  the Teichmüller character CITE{Wiki}.
  The images $\omega(k)$ of elements $k \in G$ are their
  Teichmüller representatives in $\mathbb{Z}_p$.

Parameters:
  p:
    type: Z
    constraints: prime
    
  k:
    type: Z
    constraints: 
    - > 
      $1 \leq k < p$ for $p>2$
    - >
      $k = \pm 1$ for $p=2$
    
Comments:
  comment-roots-of-unity: >
    The set of Teichmüller representatives in $\mathbb{Z}_p$
    equals the set of non-zero roots of unity in $\mathbb{Z}_p$.
    The $k$'th Teichmüller representative reduces to $k$ modulo $p$.
  
Formulas:

Programs:

References:

Links:
  Wiki:
    title: "Wikipedia: Teichmüller character"
    url: https://en.wikipedia.org/wiki/Teichm%C3%BCller_character

Similar tables:
  
Keywords:
  
Tags:
- p-adic
- zero

Data properties:
  type: Qp
  complete: no

Display properties:
  number-header: Teichmüller representative of $k$ in $\mathbb{Z}_p$

Numbers: INPUT{numbers.yaml}

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Teichmüller representatives in $\mathbb{Z}_p$

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Numbers

$p$

$k$

Teichmüller representative of $k$ in $\mathbb{Z}_p$

INPUT{numbers.yaml} (not shown in preview)

Definition

Let $p$ be a rational prime, let $q=p$ for $p>2$ and $q=4$ for $p=2$, let $G = (\mathbb{Z}/q\mathbb{Z})^\times$, and let $\omega: G \to \mathbb{Z}_p^*$ be the Teichmüller character [1]. The images $\omega(k)$ of elements $k \in G$ are their Teichmüller representatives in $\mathbb{Z}_p$.

Parameters

$p$

— integer (prime)

$k$

— integer ($1 \leq k < p$ for $p>2$
, $k = \pm 1$ for $p=2$
)

Comments

(1)

The set of Teichmüller representatives in $\mathbb{Z}_p$ equals the set of non-zero roots of unity in $\mathbb{Z}_p$. The $k$'th Teichmüller representative reduces to $k$ modulo $p$.

Links

[1]

Wikipedia: Teichmüller character

Data properties

Entries are of type: p-adic number

Table is complete: no