Question

Show that if any Gaussian prime \(p\) divides the product \(a b c \cdots\) of Gaussian primes, then \(p\) must equal one of those primes, or one of them multiplied by a unit. (Hint: Take norms of both sides.)

Short answer

Question: Show that if a Gaussian prime \(p\) divides the product of Gaussian primes \(abc\cdots\), then \(p\) must equal one of those primes or one of them multiplied by a unit. Answer: Using the properties of norms and Gaussian primes, we showed that if \(p\) divides the product of Gaussian primes, then its norm must be equal to one of their norms. Since norms are unique for each Gaussian integer, \(p\) must equal one of those primes multiplied by a unit. Therefore, if a Gaussian prime \(p\) divides the product \(abc\cdots\) of Gaussian primes, it must equal one of those primes or one of them multiplied by a unit.

Step-by-step solution

Assumptions

Assume that \(p\) is a Gaussian prime that divides the product of Gaussian primes, i.e., \(p\) divides \(abc\dots\). Therefore, we have: \(p\mid abc\cdots\)

Using Norms

Since \(p\) divides the product, this implies that \(p\) is a factor of \(N(p)\). By the properties of norms, we can take the norms on both sides: \(N(p) \mid N(abc\cdots)\)

Multiplicativity of Norms

Using the multiplicative property of norms, we can now write the right-hand side as a product of norms: \(N(p) \mid N(a)N(b)N(c)\cdots\)

Norms of Gaussian Primes

Since \(p\) is a Gaussian prime and we know that a Gaussian prime has exactly four divisors (itself, its associates, and \(\pm1\)), the only way for \(N(p)\) to divide the product of the norms of the Gaussian primes is for \(N(p)\) to be equal to one of those norms: \(N(p) = N(w)\) where \(w\) is one of the Gaussian primes in the product \(abc\cdots\) (i, with a possible multiplication by a unit).

Conclusion

Since the norms are equal, and the norms are unique for each Gaussian integer, we can safely conclude that \(p\) must equal one of those Gaussian primes multiplied by a unit: \(p = uw\) Thus, we have shown that if any Gaussian prime \(p\) divides the product \(abc\dots\) of Gaussian primes, then \(p\) must equal one of those primes, or one of them multiplied by a unit.

Key concepts

Divisibility in Complex Numbers

Understanding the concept of divisibility within the realm of complex numbers, particularly Gaussian integers, is pivotal when studying various number theory applications. In simple terms, one Gaussian integer, say 'a', is considered to be divisible by another, say 'b', if there is a Gaussian integer 'c' such that the product of 'b' and 'c' yields 'a'. This is denoted as \( b \mid a \).

We often rely on divisibility to reason about the factorization of complex numbers into their prime components. Unique to Gaussian integers, every prime element can be thought of as an indivisible unit, except by itself or its associates, which are numbers that differ only by multiplication with a unit (\( \text{i.e.,} \pm 1, \pm i \)). This characteristic allows us to explore prime factorization in similar ways to the integers but with the additional richness that complex numbers provide.

Norms in Number Theory

The concept of norms plays a crucial role in number theory. When it comes to complex numbers, particularly Gaussian integers, the norm is defined by taking a number, say \( z = a + bi \), and calculating \( N(z) = a^2 + b^2 \). What is remarkable about this computation is that it translates a complex number into a non-negative integer, which is profoundly useful when examining properties such as divisibility and prime factorization.

Since norms output integers, this property allows us to apply our understanding of divisibility and primes from the integers to Gaussian integers. Additionally, the norm is absolute in that it is always non-negative, which means it also provides us with a measure of the 'size' of a complex number.

Multiplicative Property of Norms

The multiplicative property of norms is a vital concept when dealing with Gaussian integers. It states that the norm of a product of numbers is equal to the product of their norms, formally written as \( N(ab) = N(a)N(b) \) for any Gaussian integers 'a' and 'b'.

This property holds immense importance as it preserves the structure of divisibility and primes in Gaussian integers, just as multiplication does in the realm of natural numbers. Due to this attribute, when a Gaussian prime divides a product, we can infer that its norm divides the product of the norms. This foundational principle is key to understanding the interaction between Gaussian primes and divisibility in complex numbers.

Gaussian Integers

Gaussian integers are a unique set of complex numbers that encompass all numbers of the form \( a + bi \), where both 'a' and 'b' are integers. They form a grid on the complex plane and have a rich algebraic structure, including notions of addition, subtraction, multiplication, and divisibility. Furthermore, every Gaussian integer has a norm, as previously discussed.

The prime elements within the realm of Gaussian integers, known as Gaussian primes, resemble prime numbers within the integers but they exhibit unique properties due to the two-dimensional nature of the complex plane. Gaussian primes serve as building blocks for Gaussian integers, much like prime numbers do for natural numbers, maintaining a structure called a unique factorization domain, which ensures each Gaussian integer has a unique prime factorization, up to ordering and multiplication by units (\( \pm 1, \pm i \)).