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Chapter 9: Problem 13

Sei \(p\) eine Primzahl. Bestimmen Sie \(\left|\mathbf{Z}_{p}{ }^{*}\right|\) und \(\left|\mathbf{Z}_{p^{2}}^{*}\right|\)

Short Answer

Expert verified
\( \left|\mathbf{Z}_{p}^{*}\right| = p-1 \) and \( \left|\mathbf{Z}_{p^{2}}^{*}\right| = p(p-1) \).

Step by step solution

01

Understanding the Problem

We need to find the number of elements in the multiplicative group of integers modulo a prime number \( p \) and the multiplicative group of integers modulo \( p^2 \), denoted by \( \left|\mathbf{Z}_{p}^{*}\right| \) and \( \left|\mathbf{Z}_{p^{2}}^{*}\right| \) respectively.
02

Number of Units modulo a Prime

Since \( p \) is a prime number, the multiplicative group \( \mathbf{Z}_{p}^{*} \) consists of all non-zero integers modulo \( p \). Therefore, there are \( p-1 \) such integers (i.e., 1 through \( p-1 \)). Thus, \( \left|\mathbf{Z}_{p}^{*}\right| = p-1 \).
03

Number of Units modulo a Prime Squared

For the group \( \mathbf{Z}_{p^2}^{*} \), we use Euler's totient function \( \varphi \). The formula for \( \varphi(n) \) is \( n \left( 1 - \frac{1}{p} \right) \) when \( n = p^k \). So for \( n = p^2 \), we have \( \varphi(p^2) = p^2 - p = p(p-1) \).
04

Combine Results

We have found \( \left|\mathbf{Z}_{p}^{*}\right| = p-1 \) and \( \left|\mathbf{Z}_{p^{2}}^{*}\right| = p(p-1) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Prime numbers are the building blocks of numbers and play a crucial role in various branches of mathematics, including number theory and cryptography. Here are a few key features of prime numbers:
  • A prime number cannot be divided evenly by any other number apart from 1 and itself. For example, 7 is a prime number because it cannot be divided by any other number except 1 and 7.
  • The number 2 is the smallest and the only even prime number. Every other even number can be divided by 2, and hence, cannot be prime.
  • Integers larger than 2 can either be prime or composite (i.e., they have divisors other than 1 and itself). For instance, 9 is not a prime number because it can be divided by 3.
Understanding prime numbers is important because their properties simplify complex number theories, especially when combined with other mathematical concepts. In the context of multiplicative groups, prime numbers help define the number of elements in a group of integers modulo that prime.
Euler's Totient Function
Euler's Totient Function, denoted as \( \varphi(n) \), is a crucial concept in number theory that counts the number of integers up to a given integer \( n \) that are relatively prime to \( n \). Here's how it works:
  • For a prime number \( p \), the function is straightforward: \( \varphi(p) = p - 1 \). This is because every integer from 1 to \( p-1 \) is relatively prime to \( p \).
  • For a power of a prime, say \( n = p^k \), Euler's formula gives the number of units as \( \varphi(p^k) = p^k \left( 1 - \frac{1}{p} \right) = p^k - p^{k-1} \).
  • When \( k = 2 \), this simplifies to \( \varphi(p^2) = p^2 - p \), which means there are \( p(p-1) \) numbers less than \( p^2 \) that are coprime with \( p^2 \).
Euler鈥檚 Totient Function is particularly useful in modular arithmetic and cryptography, including RSA encryption. When determining the size of a multiplicative group modulo a power of a prime, Euler's function provides an essential tool for computing the number of group elements.
Multiplicative Group
In modular arithmetic, a multiplicative group is composed of numbers that have a multiplicative inverse modulo some number \( n \). Understanding this concept involves recognizing two main points:
  • The group \( \mathbf{Z}_n^{*} \) consists of all integers less than \( n \) which are relatively prime to \( n \). These integers can multiply and result in another group member.
  • For a prime number \( p \), the multiplicative group \( \mathbf{Z}_p^{*} \) contains \( p-1 \) elements, as all numbers except 0 are divisible only by themselves and 1. Thus, each non-zero element has a modular inverse.
  • In a group modulo \( p^2 \), which is the square of a prime, Euler's Totient Function determines the size of the group as \( p(p-1) \). This includes all integers less than \( p^2 \) that are not divisible by \( p \).
The concept of multiplicative groups is vital in various branches of mathematics and applications like cryptography. They provide a framework for solving equations and constructing algorithms that rely on the properties of modular inverses and coprime relations.

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Most popular questions from this chapter

Sei \(G:=\mathbf{R} \backslash\\{-1\\}\). Wir definieren auf \(G\) eine Operation * wie folgt: \(x^{*} y:=x y+x+y\). (a) Zeigen Sie, dass \(\left(G_{2}{ }^{*}\right)\) eine Gruppe ist. (b) Geben Sie einen Isomorphismus von \((G,) \operatorname{auf}\left(\mathbf{R}^{*}, \cdot\right\), an.

Zeigen Sie, dass die Menge \(S L(n, q)\) der \(\mathrm{n} \times \mathrm{n}\)-Matrizen \(M\) 眉ber \(G F(q)\) mit \(\operatorname{det}(M)=1\) eine Untergruppe von \(G L(n, q)\) ist. Bestimmen Sie den Index dieser Untergruppe in \(G L(n, q)\)

Bestimmen Sie die Symmetriegruppe eines Rechtecks, das kein Quadrat ist. 脷berzeugen Sie sich durch eine Verkn眉pfungstabelle, dass die von Ihnen gefundenen Abbildungen wirklich eine Gruppe bilden.

Machen Sie sich klar, dass \(\mathbf{Z}_{n}{ }^{*}\) bez眉glich der Multiplikation modulo \(n\) eine Gruppe bildet. Machen Sie sich die Aussage zun盲chst am Beispiel \(n=15\) klar.

Mit \(G L(n, q)\) bezeichnen wir die Gruppe der invertierbaren \(\mathrm{n} \times \mathrm{n}\)-Matrizen mit Elementen aus \(G F(q)\). Bestimmen Sie die Ordnung von \(G L(n, q)\) f眉r \(n=2\) und \(n=3\). Zusatzfrage: K枚nnen Sie eine allgemeine Formel f眉r \(|G L(n, q)|\) angeben?
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