Carl Friedrich Gauss was the first mathematician to understand the importance of unique factorization in relation to algebraic integers. While unique factorization was a known property of integers since the time of Euclid, it is not true that it is necessarily a property of all algebraic integers.
Unique factorization is the idea that a given whole number consists of a unique set of primes. For every set of unique primes, there is one and only one number that can be built. For every number, there is one and only one unique set of primes.
Before Gauss, this idea was probably not interesting to mathematicians because it is so clearly true in the case of integers. On the other hand, when Leonhard Euler presented his proof for Fermat's Last Theorem n=3, he introduced a new set of integers in the form of a + b√-3. This was a brilliant proof and yet he made a mistake. He assumed that unique factorization automatically applied. It was Gauss's insight that this assumption requires proof.
Gauss took the idea of unique factorization very seriously when he proposed what are today called Gaussian Integers. Using i-notation, where i is the square root of -1, a Gaussian integer is defined by the set of all values that can be created by a + bi where a,b are integers. It is also represented by the form Z[i]. Using this notation, Euler's proof used integers of the form Z[√-3].
The interesting idea behind Gaussian Integers is that any theorem which is true of all Gaussian Integers is also true of all integers since integers are Gaussian Integers wheren b = 0.
But how does one reason with Gaussian Integers? On the surface, it doesn't look like they will be offer any insights beyond the standard integers. In fact, it seems, on the surface, that reasoning about them will be more difficult since they themselves are more complicated. Using standard integers, we know that 5 is a prime. But in the realm of Gaussian Integers, it is not a prime since
5 = (2 + i)(2 - i) = 4 + 1 = 5.
Gaussian Integers are interesting because they allow us to factor equations in interesting ways. For example, the equation x4 + y4 can now be factored into (x2 - y2i)(x2 + y2i). Without Gaussian Integers, there wouldn't be an easy factoring. This was really Euler's motivation for extending the integers since he was able to break x3 + y3 into a form such as:
(x + y)(x + ay)(x + by).
Coming up with proofs about Gaussian integers becomes interesting with the introduction of a special function called the norm. The norm is a mapping function that maps any Gaussian Integer to a standard integer. The norm is important because it enables to us to define Gaussian primes and because it then enables us to establish unique factorization for Gaussian Integers.
How does one find a mapping function that is guaranteed to convert any Gaussian Integer to a standard integer? The answer comes from the following mathematical equation:
(a + b)(a - b) = aa -ab +ab - bb = a2 - b2
In the case of i, this becomes:
(a + bi)(a - bi) = a2 - (b2)(-1) = a2 + b2.
The form a - bi is called the conjugate of the Gaussian Integer. Hardy and Wright (see reference below) define the norm as the product of a Gaussian Integer with its conjugate. For purposes of my discussion on Gaussian integers, I will use the Hardy and Wright definition (I should point out that this is not the standard definition of norm, see here; I will revisit this point at a later time). The important point behind the norm is that it is a mapping between the Gaussian integers and the rational integers that allows us to reason about the Gaussian integers (see here for an example of using the norm for Gaussian integers to establish a division algorithm for Gaussian integers.)
For example, the norm of 5i is (0 + 5i)(0 - 5i) = 25. Interestingly, the norm of -5i is also 25. This is important because it shows that the norm is not a one-to-one mapping. All Gaussian integers and their conjugates share the same norm.
In summary, I offer the following definitions for Gaussian Integers:
Definition 1: Conjugate of a Gaussian Integer
The conjugate of a Gaussian Integer a+bi is a-bi and likewise, the conjugate of a Gaussian Integer a-bi is a+bi.
Definition 2: Norm of a Gaussian Integer: N(α) = α * α
Following Hardy and Write (see article by Eric Weisstein for details), the norm of a Gaussian integer α is α multiplied by its conjugate α, that is, the norm N(α) = α * α.
Over the next set of blogs, I will show how norms allow us to establish unique factorization for Gaussian Integers. My next blog will cover some basic ideas about Gaussian Integer norms.