In our previous installment of the series on algebraic number theory, we took a little detour into Diophantine equations in order to provide some motivation for the theory itself. Prior to that, we had looked at different types of numbers, to give a perspective on the sorts of objects the theory deals with.
In those discussions, we touched (explicitly or implicitly) on abstract algebraic structures called groups, rings, and fields. All three of these concepts are incredibly important in the theory of algebraic numbers, and to a very large extent in the rest of modern mathematics as well. Today we’ll deal with groups and rings. In the next installment we’ll take up fields and Galois theory. This won’t be in any great detail – just the basic concepts. Further depth will be introduced later, as it becomes necessary.
We begin with groups. As with most other sorts of algebraic systems, groups are defined abstractly in terms of sets of elements satisfying certain axioms. The axioms for a group are not the simplest that an interesting mathematical system can have — monoids and semigroups have somewhat weaker axioms. But groups are just about the simplest objects that occur commonly in algebraic number theory.
A group G is a mathematical system consisting of a set of elements and one operation between any two elements of the set. If “∘” denotes the operation, in a group there are three requirements:
- the operation should be associative: x∘(y∘z) = (x∘y)∘z for all x, y, z in G;
- there should be an identity element “e”: e∘x = x∘e = x for all x in G;
- every element of G should have an inverse: x∘x-1 = x-1∘x = e.
These axioms can be stated in slightly different ways, but we don’t need to get into that.
Note one respect in which these group rules are different from the usual rules of either addition or multiplication in arithmetic: the commutative property x∘y = y∘x is not required for elements of a group, though it might hold for some or even all elements. If it does hold for all group elements, the group is said to be commutative or abelian (after Niels Abel). In the theory of algebraic numbers, whenever groups consist of actual algebraic numbers they will necessarily be commutative, since the rules of arithmentic (both addition and multiplication) still hold. But we will encounter groups that are defined in different ways that definitely won’t be commutative. Some of the hardest problems of the theory, in fact, occur in the non-commutative cases.
For a nontrivial example of a commutative group that’s important in algebraic number theory, just look at the set of all units, as we discussed in reference to Pell’s equation. As you recall, we denoted by ℤ[√n] the set of numbers of the form a+b√n, where a and b are integers, and n is a positive integer that’s not a perfect square. (ℤ[√n] is in fact a ring, as we’ll define the term in a moment.)
Within that set, consider the subset of numbers such that the equation a2 – nb2 = ±1 holds. In other words, the “norm” of a+b√n, N(a+b√n), as defined by the left hand side of the equation, has the value ±1. We noted that this condition is necessary and sufficient for a number in the subset to have a multiplicative inverse. We called such numbers units of the ring ℤ[√n]. Note that 0 is not a unit, but 1 is, and that the existence of a multiplicative inverse of any unit makes the set of units into a commutative group under multiplication (with ordinary addition being irrelevant in this group – indeed, the sums and differences of units are not units).
Another thing to note is that the requirement for an identity element is a requirement for a solution to a certain simple equation, and we have seen this in action several times. For instance, the natural numbers ℕ (non-negative integers, including 0) do not form a group under addition, because there is in general no solution to an equation of the form x+a = 0 with arbitrary a∈ℕ. But if we extend ℕ to the integers ℤ by “adjoining” all negative numbers, we have in effect simply included all formal solutions of equations x+a=0 for each a∈ℕ and gotten lucky in that the enlarged domain satisfies the group axioms without difficulty.
Very much the same thing happened with respect to the operation of multiplication when we passed from the integers ℤ to the rationals ℚ. Again, with respect to multiplication, ℤ satisfies the group axioms except for the existence of inverses. That is, we are not able to solve the equation xa = 1 for arbitrary a∈ℤ. In fact, a solution exists only for a = ± 1. But in defining the rational numbers ℚ in effect we just formally adjoined the inverses (reciprocals) 1/a for each a∈ℤ (except a=0). The resulting group with the operation of multiplication consists of all nonzero elements of ℚ. This group is sometimes denoted by ℚ×.
If we wanted to preserve the additive structure of ℤ at the same time as providing multiplicative inverses, in order to construct the ring ℚ, we would need to have been a little subtler. This process is a standard one. It is called constructing a field of fractions, and we will come back to it.
Let’s look at rings next, since they are the next major level up in axiom complexity. A ring is a mathematical system which has two distinct binary operations: “+” and “×”, which are intended to be rather like the addition and multiplication of ordinary arithmetic. If R is a ring, then it satisfies the axioms for a commutative group with respect to addition. With respect to multiplication, R must satisfy the associative axiom. Sometimes rings are not required to have a multiplicative identity element, but most in fact do. Inverse elements, however, do not typically exist, even if there is a multiplicative identity element. Addition in a ring is always commutative, but multiplication need not be. If the multiplication is commutitive, the ring is a commutative ring. The rings that occur in algebraic number theory are commutative rings if they consist of ordinary algebraic numbers, but a few important cases of rings (matrix rings for example) aren’t commutative.
In addition to the requirements on the operations of addition and multiplication seprately, they must satisfy a compatibility condition, known as the distributive law of multiplication with respect to addition:
- for all a, b, c in R, a(b + c) = ab + ac
If multiplication in R isn’t commutative, the same thing must hold for multiplication on the right as well. One consequence of this axiom is that if 0 is the additive identity element, a0=0a=0 for all a∈R.
The integers ℤ are the most obvious example of a ring. For any field F containing ℚ, there is also the concept of a ring of integers of the field F. This ring is a direct generalization of ℤ, and it is one of the central objects of study. One wants to know as much as possible about the structure of such rings, because this knowledge has extensive practical application to the study of Diophantine equations, as we shall see. Elements of a such a ring of integers are called, simply, algebraic integers.
In many respects, of groups, rings, and fields, it is rings which are most interesting. They have the complexity due to possessing two operations, but the freedom of a less restrictive set of axioms than fields. This results in many more special situations, though not all of the strong theorems about fields (such as Galois theory) apply to rings.
We’ll have a lot more to say about abstract ring theory, but in the next installment the theory of fields and Galois theory will be reviewed.