Problem 44 - A Computational Introduction to Number Theory and Algebra

Question

In Exercise 16.41, we saw that 2 factors as \(-i(1+i)^{2}\) in \(\mathbb{Z}[i]\), 16.9 Unique factorization domains (*) where \(1+i\) is irreducible. This exercise examines the factorization in \(\mathbb{Z}[i]\) of prime numbers \(p>2\). Show that: (a) for every irreducible \(\pi \in \mathbb{Z}[i],\) there exists a unique prime number \(p\) such that \(\pi\) divides \(p\); (b) for all prime numbers \(p \equiv 1(\bmod 4),\) we have \(p=\pi \bar{\pi},\) where \(\pi \in \mathbb{Z}[i]\) is irreducible, and the complex conjugate \(\bar{\pi}\) of \(\pi\) is also irreducible and not associate to \(\pi\); (c) all prime numbers \(p \equiv 3(\bmod 4)\) are irreducible in \(\mathbb{Z}[i]\).

Step-by-Step Solution

Answer

In summary, factorization in the Gaussian integers involves the following properties: (a) For every irreducible element in the Gaussian integers, there exists a unique prime number such that the irreducible element divides that prime number. (b) For prime numbers that are congruent to 1 modulo 4, they can be factored as the product of two conjugate irreducible Gaussian integers. (c) For prime numbers congruent to 3 modulo 4, they remain irreducible in the Gaussian integers.

Step 1: Proof of part (a) #

Let \(\pi \in \mathbb{Z}[i]\) be an irreducible element. We can define the norm function in \(\mathbb{Z}[i]\) as \(N(a+bi) = (a+bi)(a-bi) = a^2 + b^2\). The norm is multiplicative, meaning \(N(\alpha \beta) = N(\alpha)N(\beta)\) for any \(\alpha, \beta \in \mathbb{Z}[i]\). Let \(\pi = a+bi\) be the irreducible element. Since \(\pi\) is not a unit, we have that \(N(\pi) = a^2 + b^2 > 1\). Since \(\pi\) is irreducible, any factorization of \(N(\pi)\) in \(\mathbb{Z}\) cannot be non-trivial without one of the factors being divisible by \(\pi\), which would make it a common factor. Therefore, there exists a unique prime number \(p\) such that \(\pi\) divides \(p\). In other words, \(p = N(\pi) = a^2 + b^2\).

Step 2: Proof of part (b) #

Suppose \(p \equiv 1(\bmod 4)\) is a prime number. Then \(p = 4n+1\) for some integer \(n\). We can write \(p\) as the sum of two squares: \(p = a^2 + b^2\). By the Gaussian integers properties, \(p\) can be factored as \((a+bi)(a-bi)\). Let \(\pi = a+bi\), then \(\bar{\pi} = a-bi\). Note that \(\pi\) and \(\bar{\pi}\) are not associates since their imaginary parts have different signs. Now, we need to show that both \(\pi\) and \(\bar{\pi}\) are irreducible. Suppose there exists a non-trivial factorization of \(\pi\). Then, there must be two elements \(\alpha\) and \(\beta\), not units, such that \(\pi = \alpha \beta\). Taking norms, we have \(N(\pi) = N(\alpha) N(\beta)\). Since \(\alpha, \beta\) are not units, both \(N(\alpha), N(\beta) > 1\). Thus, \(N(\pi) = p = N(\alpha) N(\beta) \Rightarrow p\) is not prime in \(\mathbb{Z}\), which is a contradiction. Therefore, both \(\pi\) and \(\bar{\pi}\) are irreducible and \(p = \pi \bar{\pi}\).

Step 3: Proof of part (c) #

Suppose \(p \equiv 3(\bmod 4)\) is a prime number. Let \(\pi\) be an irreducible factor of \(p\) in \(\mathbb{Z}[i]\). Then, there exists another irreducible element \(\delta\), such that \(p = \pi \delta\). Taking norms, we have \(N(p) = N(\pi) N(\delta)\). Since \(N(\pi), N(\delta) > 1\), the only possible factorization is \(N(\pi) = p\) and \(N(\delta) = 1\). But this would imply that \(\delta\) is a unit, which contradicts our assumption that \(\delta\) is irreducible. Thus, \(p\) cannot be factored in \(\mathbb{Z}[i]\) and is therefore irreducible. In conclusion, we have shown that: (a) For every irreducible \(\pi \in \mathbb{Z}[i]\), there exists a unique prime number \(p\) such that \(\pi\) divides \(p\). (b) For all prime numbers \(p \equiv 1(\bmod 4)\), we have \(p=\pi \bar{\pi}\), where \(\pi \in \mathbb{Z}[i]\) is irreducible, and the complex conjugate \(\bar{\pi}\) of \(\pi\) is also irreducible and not associate to \(\pi\). (c) All prime numbers \(p \equiv 3(\bmod 4)\) are irreducible in \(\mathbb{Z}[i]\).

Key Concepts

Gaussian Integers

The concept of Gaussian integers, denoted as \( \mathbb{Z}[i] \), is a fascinating extension of the usual integer domain. These are complex numbers where the real and imaginary parts are both integers.
For example, a Gaussian integer could be expressed as \( a + bi \), where \( a \) and \( b \) are ordinary integers.

• The set \( \mathbb{Z}[i] \) forms a domain, meaning it follows certain algebraic rules, similar to normal integers.
• It includes the factorization properties, allowing us to apply the concept of "divisibility" within this set.

To explore Gaussian integers further, consider the *norm* of a Gaussian integer, defined as \( N(a+bi) = a^2 + b^2 \). This norm helps to understand their size and properties.
Even though they appear exotic, Gaussian integers share many properties with the regular integers, including unique factorization, which leads us to the application of prime numbers within this domain.

Prime Numbers

Prime numbers are the building blocks of numbers in mathematics. In the realm of Gaussian integers, the idea of a prime number is slightly nuanced but remains crucial.
A prime in ordinary integers is only divisible by 1 and itself, but in \( \mathbb{Z}[i] \), prime behavior changes based on certain conditions.

• If a regular prime number is of the form \( p \equiv 1 \pmod{4} \), it can be expressed as a product of two Gaussian integers: \( p = \pi \bar{\pi} \).
• For primes of the form \( p \equiv 3 \pmod{4} \), these remain primes even in \( \mathbb{Z}[i] \).

This distinction is fascinating because it highlights how numbers factor differently depending on the mathematical system used. Thus, the exploration of prime numbers within Gaussian integers provides a deeper insight into how they intertwine with deeper algebraic systems.

Irreducibility

Irreducibility in the context of Gaussian integers plays a similar role as prime numbers in regular integers. An element in \( \mathbb{Z}[i] \) is considered irreducible if it cannot be expressed as a product of two non-unit elements.
This concept is essential to understand how elements function as the "primes" of this set.

• An element \( \pi = a+bi \) is irreducible if its norm \( N(\pi) = a^2 + b^2 \) is a prime number.
• The uniqueness of factorization is preserved in \( \mathbb{Z}[i] \), as any irreducible integer is associated with a unique prime.

Understanding irreducibility helps in proving how primes divide in \( \mathbb{Z}[i] \). If an integer's norm is a prime, it fundamentally links to the unique factorization domain properties, ensuring every integer in \( \mathbb{Z}[i] \) can be broken down uniquely into irreducible elements, paralleling how ordinary integers can be broken down into primes.