Problem 4 - A Book of Abstract Algebra

Question

If \(p\) is a prime, \(\phi\left(p^{n}\right)=p^{n}-p^{n-1}=p^{n-1}(p-1)\). (HINT: For any integer \(a, a\) and \(p^{n}\) have a common divisor \(\neq \pm 1\) iff \(a\) is a multiple of \(p .\) There are exactly \(p^{n-1}\) multiples of \(p\) between 1 and \(p^{n}\).)

Step-by-Step Solution

Answer

The totient function for \(p^n\) is \(\phi(p^n) = p^{n-1}(p-1)\).

Step 1: Understand Euler's Totient Function

Euler's Totient Function, denoted as \(\phi(n)\), counts the number of positive integers up to \( n \) that are relatively prime to \( n \). Two numbers are relatively prime if their greatest common divisor (GCD) is 1.

Step 2: Recognize the Prime Power Property

The function \(\phi(p^n)\) for a prime power \(p^n\) can be computed using the formula \(\phi(p^n) = p^n - p^{n-1}\). This is because the only numbers between 1 and \(p^n\) that are not relatively prime to \(p^n\) are those that are multiples of \(p\).

Step 3: Calculate the Number of Non-Relatively Prime Numbers

In the range from 1 to \(p^n\), any number that is a multiple of \(p\) will not be relatively prime to \(p^n\). These numbers form a sequence: \(p, 2p, 3p, \ldots, p^n\). There are exactly \(p^{n-1}\) such multiples.

Step 4: Derive the Totient Function Formula

Subtract the number of non-relatively prime integers from the total count \(p^n\). Thus, \(\phi(p^n) = p^n - p^{n-1}\). Factor this expression to highlight the structure in the hint: \(\phi(p^n) = p^{n-1}(p-1)\). This confirms the formula definition given in the exercise.

Key Concepts

Prime Numbers

Prime numbers are unique because they have no divisors other than 1 and themselves. In simpler terms, a prime number is any natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Some examples of prime numbers include:

• 2
• 3
• 5
• 7
• 11

Notice that all of these numbers can only be divided evenly by the number 1 and themselves.
Prime numbers play a fundamental role in various fields of mathematics, including number theory and cryptography. They serve as the basic building blocks from which all other natural numbers can be constructed, known as the fundamental theorem of arithmetic. This theorem states that every integer greater than 1 is either a prime number itself or can be factored into prime numbers.
Understanding prime numbers is essential when working with Euler's Totient Function, especially when it involves formulas like \(\phi(p^n)\) as seen in the exercise. This is because the properties of prime numbers simplify calculations and reveal important relationships between numbers.

Greatest Common Divisor (GCD)

The greatest common divisor, often abbreviated as GCD, is the largest number that can divide two or more numbers without leaving a remainder. It's an essential concept when determining whether two numbers are relatively prime.

To find the GCD of two numbers, you can use several methods, such as the Euclidean algorithm:

• Start with two numbers, divide the larger by the smaller, and find the remainder.
• Then replace the larger number with the smaller number and the smaller number with the remainder.
• Repeat this process until the remainder is zero.
• The last non-zero remainder is the GCD.

For example, to find the GCD of 48 and 18:

• Divide 48 by 18 to get a remainder of 12.
• Next divide 18 by 12 to get a remainder of 6.
• Then, divide 12 by 6 to get a remainder of 0, so the GCD is 6.

In the context of Euler's Totient Function, understanding the GCD helps us identify numbers that are relatively prime to a given number, which is crucial for accurate calculations.

Relatively Prime Numbers

Two numbers are considered relatively prime if their greatest common divisor (GCD) is 1. This means that they do not share any divisors other than 1, even if neither number is prime itself.
Relatively prime numbers are important in various aspects of math, especially when dealing with Euler's Totient Function \(\phi(n)\), which counts the numbers up to \(n\) that are relatively prime to \(n\).
To determine if two numbers are relatively prime, you can also use their prime factorization. If the numbers share no common factors (other than 1), they are relatively prime.

Here is a simple example: Consider the numbers 8 and 15. Their factorizations are:

• 8 = 2 × 2 × 2
• 15 = 3 × 5

Since they have no common factors, 8 and 15 are relatively prime. Recognizing relatively prime numbers is crucial when applying various mathematical functions, as it directly influences outcomes such as those calculated by Euler’s Totient Function.