An elementary treatment of algebraic number theory will typically contain four major results: first, elementary facts about number fields (in particular, the fact that their rings of integers are Dedekind domains); second, unique factorization into ideals; third, finiteness of the class group; and fourth, Dirichlet’s theorem on units. I skipped the first part because I’ve found that without knowing where it leads, it looks boring and pointless. I’m going to go back to it shortly. But now I’m going to set up parts three and four by talking about groups.
Before going any further, I should add that the old Good Math, Bad Math site dealt with group theory a little bit. But it didn’t deal with the group theory that’s useful for this sort of algebraic number theory.
You may recall what I said a hundred and something posts ago, that a group is an algebraic structure with a single binary operation, which we write as *, which satisfies the following conditions:
1. Associativity: for all a, b, and c in G, (a*b)*c = a*(b*c)
2. Identity: there exists an element 1 such that for all a in G, 1*a = a*1 = a
3. Inverses: for each a, there exists an element a^(-1) such that a*a^(-1) = a^(-1)*a = 1
In addition, it’s implied in the definition of the binary operation that we have closure – that is, for each a and b in G, a*b is in G.
I’m going to take a gigantic shortcut and use examples that have a fourth property, commutativity: a*b = b*a for all a and b. Commutative groups are called abelian groups, after Abel; abelian groups, especially finite ones, have an immensely restricted structure, which makes calculations with them relatively simple.
But anyway, we have a few examples from ring theory of abelian groups. The additive structure of every ring is an abelian group under addition. The set of units of a ring form a group under multiplication, since the product of two units is again a unit, and every unit is by definition invertible; in particular, the nonzero elements of a field form an abelian group under multiplication. The fractional ideals of a Dedekind domain form a group under multiplication.
Given a group, we can define subgroups as subsets that still form groups. For example, take Z, the group of integers under addition. The set of all even integers still forms a group, so it’s a subgroup of Z, which we write as 2Z. We can add two even integers and still get an even integer; the inverse of every even integer is still even; and the identity element, 0, is even.
In addition, given two groups, A and B, we can take their product A*B. The general element of A*B is (a, b), where a is in A and b is in B. It’s easy to verify that (a1, b1)*(a2, b2) = (a1a2, b1b2) satisfies the properties of a group, with identity element (1, 1) and inverses of the form (a^(-1), b^(-1)). If we multiply a group by itself, we can write something like A^n for any integer n.
If H is any subgroup of G, then for each g in G, we can define gH, the set of all elements of the form gh where h is in H. We call sets of this form left cosets of H (sets of the form Hg are right cosets; obviously, if G is abelian, left and right cosets are the same). Obviously, every element of G lies in some left coset of H – after all, g is in gH.
Further, if the cosets g1H and g2H have any element in common, say g1h1 = g2h2, then we have g1 = g2h2h1^(-1) = g2h3 since h2h1^(-1) is in H. Then g1H = g2h3H = g2H since h3H = H.
Now, if G is finite, then H is obviously finite. The various cosets of H obviously have the same number of elements, are disjoint (since by the previous paragraph, if they contain an element in common, they’re equal), and cover G. So the number of elements of G, |G|, is equal to the number of elements of H, |H|, times the number of different cosets of H. In other words, |H| divides |G|. This is called Lagrange’s theorem and is the first major theorem in group theory.
Finally, if H is a subgroup of G, we’re going to want to turn the cosets of H into a group. But to do that, we need to make sure left and right cosets are the same, that is, gH = Hg for all g in G. That condition is equivalent to gHg^(-1) = H for all g; if H satisfies that, it’s called normal. Clearly, if G is abelian, every H satisfies that condition. If G is not abelian, H may or may not be normal; there exist non-abelian groups all of whose subgroups are normal, and non-abelian groups with no normal subgroups, except for the trivial subgroup consisting just of the identity element and the improper subgroup consisting of the entire group.
If H is a normal subgroup of G, then we can try defining the quotient group G/H to be the group of cosets of H, with g1H * g2H = g1g2H. Since H is normal, g1h1*g2h2 is indeed in g1g2H: g2H = Hg2, so we can write g2h2 as h3g2, and then (h1h3)g2 as g2h4, so we get g1g2h4, which is in g1g2H. On the other hand, every element of g1g2H can be written as g1g2h1 = (g1*1)*(g2h1) which is in g1H * g2H. What that means is that if we replace g1 by g1h, multiplying by g2H will still give us the same coset.
The main reason to care about groups is that they feature prominently in ideal theory. If R is a Dedekind domain with fraction field K, then K*, the set of all nonzero elements of K, forms a group under multiplication. In addition, R*, the group of units of R, is a subgroup of K*. Further, F(R), the fractional ideals of R, form a group, and P(R), the principal ideals of R, form a subgroup of F(R).
Since all groups here are abelian, all subgroups are normal. So we can take the quotient group F/P, always denoted H. If R is a PID, then all ideals are principal, so F = P, and H has just one element, P. If R is not a PID, it will have more elements – for example, if R = Z[SQRT(-5)], then H has two elements, P and (2, 1+SQRT(-5))P. So the bigger H is, the further away R is from being a PID. The relevant result here is that H is always finite.
Now, if you look at it the right way, you’ll see that P = K*/R*. In P, (a) = (b) iff (a/b) = (1), iff a/b is in R*, iff aR* = bR*. R* can create problems just like H – remember that in my first example of algebraic number theory, involving x^2 – 2 = y^3, the problem involved stray units, not non-principal ideals. The relevant result here is Dirichlet’s theorem on units, which says that R* = Z^m * Z/nZ; Z/nZ is simply the group of roots of unity, which includes at least 2 elements, 1 and -1, and m is a constant defined by certain properties of K that I’ll explain later.