A Gaussian factor is a Gaussian integer that divides another Gaussian integer. In the context of factorization, these are the fundamental components – specifically, Gaussian primes and units – that multiply together to form a Gaussian composite number.
Understanding Gaussian Integers
Before defining Gaussian factors, it's essential to understand what a Gaussian integer is. A Gaussian integer is a complex number of the form a + bi, where a and b are ordinary integers.
According to the provided reference:
A Gaussian integer is either the zero (0), one of the four units (±1, ±i), a Gaussian prime, or composite.
- Zero: The number 0.
- Units: The Gaussian integers 1, -1, i, and -i. These are analogous to ±1 in ordinary integers – they divide every Gaussian integer.
- Gaussian Prime: A non-unit Gaussian integer z that cannot be written as a product z = w k where w and k are non-unit Gaussian integers. For example, 1+i and 2+i* are Gaussian primes.
- Gaussian Composite: A non-unit Gaussian integer that can be written as a product of two non-unit Gaussian integers. For example, 2 is composite in Gaussian integers because 2 = (1+i)(1-i).
What are Gaussian Factors?
As mentioned, a Gaussian factor of a Gaussian integer z is any Gaussian integer w such that z can be expressed as z = w k, where k is also a Gaussian integer. This is similar to how factors work with ordinary integers (e.g., 3 is a factor of 6 because 6 = 3 2).
However, the term "Gaussian factor" is most commonly used in the context of factorization. The reference describes a table showing Gaussian integers along with their "explicit factorization" or labeled as "(p)" if prime. This factorization breaks down a composite Gaussian integer into a product of simpler Gaussian integers.
The components of this factorization are the Gaussian factors. These factors are typically:
- Gaussian Primes: These are the irreducible building blocks in the ring of Gaussian integers, analogous to prime numbers in ordinary integers.
- Units (±1, ±i): These are always factors of any Gaussian integer but are often ignored in discussions of "prime factorization" as they don't affect divisibility in a fundamental way (multiplying by a unit just changes the "associate" of a number).
So, when we talk about the factorization of a composite Gaussian integer, its Gaussian factors are the Gaussian primes and units whose product equals the original number.
Factorization of Gaussian Integers
Just as composite ordinary integers can be factored into a unique product of prime numbers (ignoring the order and signs), composite Gaussian integers can be factored into a unique product of Gaussian primes (ignoring the order and units).
The reference's mention of a table with "explicit factorization" highlights this process.
Here are some examples of Gaussian integer factorizations and their factors:
| Gaussian Integer | Factorization | Gaussian Factors | Notes |
|---|---|---|---|
| 2 | (1+i)(1-i) | 1+i, 1-i | 1+i and 1-i are Gaussian primes. |
| 5 | (2+i)(2-i) | 2+i, 2-i | 2+i and 2-i are Gaussian primes. |
| 3 | 3 (labeled '(p)') | 3 | 3 is a Gaussian prime. |
| 4 | 2 * 2 | 2, 2 | 2 is composite (2=(1+i)(1-i)), factors can be composite. A full prime factorization would involve 1+i and 1-i factors. |
| 10 | (1+i)(1-i)(2+i)(2-i) | 1+i, 1-i, 2+i, 2-i | A complete prime factorization. |
In these examples, the terms in the "Factorization" column are the Gaussian factors that result in the original Gaussian integer.
Gaussian Units as Factors
The four units (1, -1, i, -i) are trivial Gaussian factors for any Gaussian integer. Any Gaussian integer z can be written as z = 1 z, z = -1 (-z), z = i (-i z), and z = -i (i z). These are analogous to how 1 and -1 are factors of any ordinary integer. While technically factors, they don't change the prime factorization in a meaningful way, other than changing the unit associated with the prime factors.
Gaussian Primes as Factors
Gaussian primes are the essential Gaussian factors that make up composite Gaussian integers. They are the irreducible components that cannot be broken down further into non-unit Gaussian integers. Identifying the Gaussian prime factors is the goal of prime factorization for a Gaussian integer.
In summary, a Gaussian factor is a divisor in the ring of Gaussian integers. When discussing the factorization of a Gaussian integer, the factors are the Gaussian primes and units that constitute the number.
