To expand the concept of integers, we need to understand the algebraic structures of rings and fields. This knowledge can be found online or in corresponding abstract algebra textbooks. Proposing these two algebraic structures abstractly allows us to handle the properties of different number rings and number fields in a general way. In the set of natural numbers \mathbb{N}, it is very convenient to define and compare the size of two numbers, and any subset of natural numbers has a least element. Combining these two points, we say that \mathbb{N} is “well-ordered” (which is also the basis of mathematical induction). In a well-ordered structure, the proofs of many properties become very simple, such as the Fundamental Theorem of Arithmetic. However, general number rings and number fields do not have such a “well-order”; for example, any two complex numbers cannot be compared in size. Therefore, general ideas not based on well-ordering become even more important.
Rings and Fields
For the definition of a Ring, one can refer to the Wikipedia entry on “Ring (mathematics).” Simply put, a ring refers to a set where addition and multiplication can be performed between its elements, satisfying certain necessary properties such as closure under operations and commutativity of addition. In number theory, most cases involve the study of number rings, which refer to sets of numbers where the addition and multiplication of elements are usually ordinary addition and multiplication of numbers. For example, all real integers constitute a number ring \mathbb{Z}, which is infinite; all even integers also constitute a number ring 2\mathbb{Z}; for a prime p, the set \{0, 1, 2, \dots, p-1\} under modulo p also constitutes a ring, and more specifically, it is a field.
A Field is a special case of a ring. Elements in a field can not only undergo addition and multiplication (the requirement of an additive inverse means addition also implies subtraction), but also division. One can refer to “Field (mathematics).” Under the usual four arithmetic operations, the set of real rational numbers \mathbb{Q} is the smallest number field.
Gaussian Integers
The complex numbers we usually speak of are also called Gaussian numbers. Abstractly defined, they are: \mathbb{C}=\{a+b i \mid a,b\in \mathbb{R},\ i^2=-1\} When a and b are both real integers, they are called Gaussian Integers: \mathbb{Z}[i]=\{a+b i \mid a,b\in \mathbb{Z},\ i^2=-1\} The addition and multiplication defined on \mathbb{Z}[i] are ordinary complex addition and multiplication. On this basis, Gaussian integers constitute an integral ring.
Next, we introduce the norm of Gaussian integers. The norm operation is a mapping from elements of a number ring to \mathbb{R}. Unlike \mathbb{R}, elements in a general number ring cannot be compared in size—that is, they cannot be ordered. However, to provide a sense of ordering for some elements in the number ring, the norm is defined, denoted by N (the first letter of “Norm”). It is required that the norm of the product of two elements equals the product of their norms. For Gaussian integers \mathbb{Z}[i], the norm is defined as: N(a+bi)=(a+bi)(a-bi)=a^2+b^2 Of course, norms are not exclusive to integers; general complex numbers can define a norm in the same way. Furthermore, the norm of an integer is not necessarily an integer, depending on the definition of the norm. Without considering other aspects, the norm of a complex number could also be defined as N(a+bi)=\sqrt{a^2+b^2}. However, in number theory, we usually deal with integer or rational problems, and we hope that integers or rational numbers can have an integer or rational norm to facilitate our research.
With the norm, we can define units. Units in \mathbb{Z}[i] refer to Gaussian integers with a norm of 1, which are \pm 1, \pm i. It is not difficult to find that these four numbers are all invertible, and only these four numbers are invertible; thus, units are also called invertible elements. “Invertible” means that both a+bi and \frac{1}{a+bi} belong to \mathbb{Z}[i]. If two Gaussian integers differ only by a unit factor, then these two Gaussian integers are said to be associates.
Following the concept of divisibility, if Z_1=Z_2 Z_3, where Z_1, Z_2, Z_3 \in \mathbb{Z}[i], then we say Z_2 (or Z_3) divides Z_1, denoted as Z_2|Z_1 (or Z_3|Z_1). It can be seen that Z_2|Z_1 \Rightarrow N(Z_2)|N(Z_1).
We can also define the common divisors and the greatest common divisor of two Gaussian integers. If Z_3|Z_1 and Z_3|Z_2, then Z_3 is a common divisor of Z_1 and Z_2. If for any common divisor Z_4 of Z_1 and Z_2, we have Z_4|Z_3, then Z_3 is their greatest common divisor. The greatest common divisor can be viewed as the common divisor with the largest norm. If the greatest common divisor of two numbers is a unit, then the two numbers are said to be coprime.
Finally, we can define Gaussian primes. Let Z_1 be a Gaussian integer. If for any Gaussian integers Z_2 and Z_3 that are not units, Z_1 \neq Z_2 Z_3, then Z_1 is a Gaussian prime. This is a generalization of the concept of prime numbers in real integers, usually denoted by the letter \pi. Note that in Gaussian integers, 2 is no longer a prime number because 2=(1+i)(1-i). Among them, (\pm 1 \pm i) are the four Gaussian primes with the smallest norm. For more detailed content on Gaussian integers and Gaussian primes, please refer to Chapter 9 of Lectures on Number Theory (Vol. 2) by Sun Qi and Ke Zhao.
Modulo 1+i Analysis
In real integers, we have parity analysis, where it is easy to see if a number is odd or even and make some basic feasibility judgments. However, in Gaussian integers, 2 is no longer a prime, so parity analysis loses its meaning. Nevertheless, there exists a similar “modulo 1+i analysis,” which holds a status in Gaussian integers equivalent to parity analysis in real integers. These properties play a core role in proving Fermat’s Last Theorem for n=4.
1. First is the operation regarding 1+i itself. Note that (1+i)^2=2i, (1+i)^4=-4, and (1-i)i=1+i.
2. 1+i | a+bi if and only if 2 | a+b.
Two proofs are provided here. First, let (m+ni)(1+i)=a+bi. Expanding gives (m-n)+(m+n)i=a+bi, so a+b=(m-n)+(m+n)=2m, thus 2|a+b. This proof is simple and reversible, but not very heuristic. The second proof: assume 1+i|a+bi, then it must follow that 1-i|a-bi, because the squares of \pm i are both -1, and their statuses are equal. Multiplying them gives (1+i)(1-i)|(a+bi)(a-bi). Since (1+i)(1-i)=2 and (a+bi)(a-bi)=a^2+b^2=(a+b)^2-2ab, it follows that 2|a+b. The reversibility of this is not obvious, but it is easier to generalize to other number rings.
3. If 1+i \nmid a+bi, then (a+bi)^2 \equiv \pm 1 \pmod 4.
Since 1+i \nmid a+bi, then 2 \nmid a+b, so one of a, b is odd and the other is even. Since (a+bi)^2=a^2-b^2+2abi, where 4|2abi, if a is even and b is odd, then 4|a^2 and -b^2 \equiv -1 \pmod 4; if b is even and a is odd, then 4|b^2 and a^2 \equiv 1 \pmod 4. Therefore, (a+bi)^2 \equiv \pm 1 \pmod 4. Since -4=(1+i)^4, this also implies (a+bi)^2 \equiv \pm 1 \pmod{(1+i)^4}.
4. If 1+i \nmid a+bi, then (a+bi)^4 \equiv 1 \pmod 8.
Since 1+i \nmid a+bi, then 2 \nmid a+b, so one of a, b is odd and the other is even. Since (a+bi)^4=(a^4+b^4-6a^2 b^2)+4ab(a^2-b^2)i, without loss of generality, assume a is odd and b is even. Then every term of b^4-6a^2 b^2 +4ab(a^2-b^2)i is divisible by 8, while a^4 is the fourth power of an odd number, which leaves a remainder of 1 when divided by 8. Since -8i=(1+i)^6, this also implies (a+bi)^4 \equiv 1 \pmod{(1+i)^6}.
To expand the concept of integers, we have introduced a large amount of material, much of which is conceptual, which can be painful to understand and remember. However, when we later see the concise proofs brought about by these expanded integers, we will deeply realize that this effort was worth it. But we cannot experience that joy immediately, because we are still missing one thing: the Unique Factorization Domain.
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