Question

(i) Let \(\mathbb{F}_{q}\) be a finite field with \(q\) elements. Prove Wilson's theorem \(\prod_{a \in \mathbb{F}_{q}^{\times}} a=-1\). Hint: Every \(a \in \mathbb{F}_{q}^{\times}\)different from \(\pm 1\) has \(a^{-1} \neq a\). (ii) Prove a converse of Wilson's theorem: If \(n\) is an integer such that \((n-1) ! \equiv-1 \bmod n\), then \(n\) is prime.

Short answer

(i) The product in \\( F_q^\\times\\) is \\(-1\\). (ii) If \\( (n-1)! \\equiv -1 \\ \\bmod \\ n\\), then \\(n\\) is prime.

Step-by-step solution

Understanding the Finite Field

A finite field \(F_q\) is a set with \(q\) elements where arithmetic operations (addition, subtraction, multiplication, and division) are defined, except for division by zero. The set \(F_q^\times\) represents the non-zero elements of \(F_q\).

Apply Wilson's Theorem in a Finite Field

Wilson's theorem for a finite field \(F_q\) states that the product of all non-zero elements is \(-1\). For any element \(a\) in \(F_q^\times\) distinct from \(\pm 1\), its inverse also lies in \(F_q^\times\). The product of \(a\) and its inverse \(a^{-1}\) is 1, which cancels each pair \(a, a^{-1}\). The remaining pair is \(1 \cdot (-1) = -1\).

Prove the Converse of Wilson's Theorem

Suppose \(n\) is not prime. Then, \(n = ab\) with \(1 < a < n\) and \(1 < b < n\). In \((n-1)! \), both \(a\) and \(b\) appear, causing it to be divisible by \(n\), hence \( (n-1)! \ot\equiv -1 \ \bmod \ n\). Thus, if \( (n-1)! \equiv -1 \ \bmod \ n\), \(n\) must be prime.

Key concepts

Finite Fields

Finite fields, also known as Galois fields, are mathematical structures with a finite number of elements, denoted by the symbol \(\mathbb{F}_q\). These fields are pivotal in various areas of mathematics, particularly in algebra and number theory.
Each finite field \(\mathbb{F}_q\) consists of \(q\) elements, where \(q\) is typically a power of a prime number. This ensures that every nonzero element has a multiplicative inverse. The operations within finite fields are similar to those in ordinary arithmetic (addition, subtraction, multiplication, and division except by zero).

• For example, in the finite field \(\mathbb{F}_p\), where \(p\) is a prime number, we can perform calculations using integers modulo \(p\).
• In these fields, every element can be multiplied except zero, and inverse exists for each non-zero element.

Finite fields are not only theoretical constructs; they are useful in diverse applications such as error detection and correction in coding theory, cryptographic algorithms, and computer algebra systems.

Prime Numbers

Prime numbers are the fundamental building blocks of number theory. A prime number is an integer greater than 1, which has no divisors other than 1 and itself. These numbers hold significant interest because of their unique properties and applications in various fields.

• The smallest prime numbers are 2, 3, 5, 7, and 11. Note that 2 is the only even prime number.
• Primes are closely tied to several mathematical theorems and concepts, including Wilson's Theorem.

Understanding whether a number is prime or not plays a crucial role in encryption, coding theory, and even in proving certain mathematical statements. For instance, the converse of Wilson's Theorem uses the property of prime numbers to establish a relationship involving factorial moduli.

Mathematical Proof

A mathematical proof is a demonstration that a particular statement, proposition, or theorem is necessarily true. Typically, proofs use logical reasoning and previously established results. They are essential for ensuring the validity of mathematical arguments.
Wilson's Theorem requires proving that a particular relationship holds true in finite fields. Here's a glimpse of such a proof:

• First, verify the conditions of the theorem, ensuring all assumptions are met.
• Utilize logical deductions based on known properties of finite fields and prime numbers.
• Finally, illustrate the step-by-step reasoning that leads to a conclusive proof.

The beauty of mathematical proofs lies in their ability to demonstrate truth beyond intuition or empirical observation. In number theory, proofs often reveal underlying patterns and connections that may not be immediately obvious.

Number Theory

Number theory is a branch of mathematics dedicated to exploring the properties and interrelations of numbers, particularly integers. It is a subject rich with theory and applications, influencing numerous other areas, from computer science to cryptography.
At the heart of number theory are various intriguing problems and theorems, such as Wilson's Theorem, which relates to divisibility and the properties of prime numbers.

• Number theory addresses fundamental questions about integer divisibility.
• It delves into both simple concepts like divisors and more complex ones like modular arithmetic.

Number theory provides critical insights into the nature of numbers, helping unravel complex numerical phenomena. Its principles are applied in prime testing, factoring large numbers, and developing encryption systems, among other areas. The robust connections it forms with primes and finite fields are foundational to many modern mathematical problems and solutions.