Problem 1 - A Book of Abstract Algebra

Question

If \(p\) is a prime, find \(\phi(p)\). Use this to deduce Fermat's theorem from Euler's theorem.

Step-by-Step Solution

Answer

For a prime \( p \), \( \phi(p) = p - 1 \), leading to Fermat's theorem: \( a^{p-1} \equiv 1 \pmod{p} \) for any integer \( a \) not divisible by \( p \).

Step 1: Understand Euler's Totient Function

Euler's Totient function, denoted as \( \phi(n) \), counts the number of integers up to \( n \) that are relatively prime to \( n \). In simpler terms, it checks how many numbers from 1 to \( n \) have no common factors with \( n \) other than 1.

Step 2: Calculate \(\phi(p)\) for Prime Number \(p\)

If \( p \) is a prime number, every number less than \( p \) is relatively prime to \( p \). Hence, \( \phi(p) = p - 1 \). For example, if \( p = 7 \), then the integers 1 through 6 are all relatively prime to 7, so \( \phi(7) = 6 \).

Step 3: Recall Euler's Theorem

Euler's theorem states that for any integer \( a \) that is relatively prime to \( n \), the following holds: \[ a^{\phi(n)} \equiv 1 \pmod{n} \]. This is a fundamental result in number theory.

Step 4: Deduce Fermat's Little Theorem

Since \( p \) is prime, and from Step 2 we have \( \phi(p) = p-1 \), Euler's theorem becomes \( a^{p-1} \equiv 1 \pmod{p} \) for any \( a \) such that gcd(\(a, p\)) = 1. Fermat's Little Theorem states that for any integer \( a \) not divisible by \( p \), it holds that \( a^{p-1} \equiv 1 \pmod{p} \), which is exactly Euler's result for \( n = p \).

Key Concepts

Euler's Totient Function

Euler's Totient Function, commonly denoted as \( \phi(n) \), is an essential concept in number theory. It is used to calculate how many numbers from 1 to \( n \) are relatively prime to \( n \). A number is considered relatively prime to \( n \) if it shares no common factors with \( n \) other than 1.

For example, let's say \( n = 12 \). The numbers that are relatively prime to 12 are 1, 5, 7, and 11. That makes \( \phi(12) = 4 \). If \( n \) is a prime number \( p \), calculating Euler's Totient Function becomes straightforward since all numbers less than \( p \) are relatively prime to \( p \). Thus, for a prime number \( p \), \( \phi(p) = p - 1 \).

• Key Concept: \( \phi(n) \) measures relative primeness up to \( n \).
• The totient function simplifies significantly for prime numbers.
• When \( p \) is prime, \( \phi(p) = p - 1 \).

Euler's Theorem

Euler's Theorem serves as a pivotal milestone in number theory. It states that if you have an integer \( a \) which is relatively prime to another integer \( n \), the following mathematical relationship holds: \[ a^{\phi(n)} \equiv 1 \pmod{n} \]. This essentially means that raising \( a \) to the power of the totient of \( n \) will leave you with a remainder of 1 when divided by \( n \).

To make it simpler, imagine \( a = 3 \) and \( n = 7 \). Since 3 and 7 are relatively prime, and for the prime number 7, \( \phi(7) = 6 \), it follows that \( 3^6 \equiv 1 \pmod{7} \). Euler's Theorem thus proves the relationship between exponential powers and modular arithmetic in number theory.

• Main Takeaway: The theorem links \( a^{\phi(n)} \) and 1 mod \( n \).
• Works when \( a \) and \( n \) are coprime.
• Useful for simplifying huge exponentiations.

Prime Numbers

Prime numbers form the building blocks of number theory. A prime number is an integer greater than 1 that only has 1 and itself as divisors. This implies that a prime number can't be divided evenly by integers other than 1 and itself.

The simplicity of prime numbers is deceptive. They are crucial in various mathematical domains, from simple divisibility rules to complex cryptographic systems. When dealing with prime numbers, many properties and theorems become easier to apply, as seen in the specific case of Euler's Totient function and Euler's Theorem.

For instance, if you pick any prime \( p \) and an integer \( a \) that is not a multiple of \( p \), the number \( a \) is automatically relatively prime to \( p \). This is because prime \( p \) shares no divisors with \( a \) other than 1. Vibing off of this property, both Fermat's Little Theorem and Euler's Theorem derive straightforward results when applying primes.

• Definition: An integer greater than 1 with no divisors other than 1 and itself.
• Crucial for theorems like Fermat’s Little Theorem.
• Forms the basis for understanding divisibility and arithmetic properties.