Let R be a ring. An ideal of R is a subring I of R such that if a I, r R, then a · r I and r · a I. Let R be a ring with identity 1, and let a R. If a has an inverse with respect to multiplication, then we say that a is a unit of R. A commutative ring R is an integral domain if R contains no zero divisors. In other words, R is an integral domain if the product of any two nonzero elements of R is nonzero. The nth roots of 1 in C form a cyclic group of order n. A generator of this group is n=cos(360/n)+i sin(360/n). The set of integers. The complex numbers. Let n be an integer greater than 1. Then n can be expressed as a product of primes.

### Unique Factorization and Fermat's Last Theorem

In Euclidean geometry one often meets right triangles whose sides are integers. For example, there is the famous 3,4,5 triangle. If the sides of a right triangle are x and y and the hypotenuse is z, then the Pythagorean theorem asserts that

(1)
x2 + y2 = z2

Determining all right triangles whose sides are integers is determining all integral solutions of (1) - that is, solutions for which x,y and z are integers. It can be shown that all solutions of (1) are of the form

(2)
x = c(a2 - b2), y = 2abc, z = c(a2 + b2)  a,b,c Z

It is easy to check that (2) provides a solution to (1) for every a and b. The problem of solving polynomial equations in integers goes back to the ancient Greeks, and such equations are called Diophantine equations after the Greek mathematician Diophantus of Alexandria, who first made an extensive study of them. We used the theory of congruences to solve the linear Diophantine equation

(3)
ax + by = c,     a,b,c Z.

This was not an especially difficult task. However, this is not a very representative example. For Diophantine equations are usually very difficult to solve. A given Diophantine equation may have no solutions, a finite number of solutions, or an infinite number of solutions. Equation (1) is of the latter type. The equation x2 + y2 = 1 has only four solutions: (x,y) = (1,0), (0,-1), (-1,0), (0,1); the equation x2 + y2 = -1 has no solutions.

Because of the challenges Diophantine equations present, they have been the fascination of mathematical amateurs for many centuries. One of the great amateurs was Pierre Fermat, a seventeenth century French jurist who took up mathematics as a hobby. Earlier we met one of Fermat's more famous contributions, Fermat's Little Theorem. Fermat was very interested in the solution of Diophantine equations and carefully studied the works of Diophantus. Fermat often made marginal notes in his copy of Diophantus's works. On the page on which (1) was discussed, Fermat asserted that if n > 2, then

(4)
xn + yn = zn

has no solution in integers x,y,z none of them 0. Furthermore, he claimed to have discovered "a truly marvelous demonstration which this margin is too narrow to contain." It is very doubtful that Fermat had a valid proof of this assertion, which became known as Fermat's last theorem. Despite tremendous advances in mathematics, it took nearly three hundred and fifty years to prove this statement, and even then it relied on mathematics that simply were not available to Fermat. (See Proof of Fermat's Last Theorem in the appendix.) The many attempts to prove it did lead to the creation of much beautiful and important mathematics. In fact, the modern theory of rings is the intellectual descendant of these attempts.

Before Fermat's last theorem was known to be true or false, it was proved for special cases of n. For example, Fermat had a proof for n = 4, as did Leibniz and Euler. The case n = 3 was settled by Euler, while the case for n = 5 was disposed of by Dirichlet and Legendre.

Proposition 1: It suffices to prove Fermat's last theorem for n = 4 and for n = p, where p is a prime.

Proof: Let us assume Fermat's theorem for n = 4 and n = p, and let us prove it for general n > 2. Assume that n > 2 is not prime. If n is divisible by an odd prime p, then n = pk for some k Z. If xn + yn = zn has a solution (x0,y0,z0), x0, y0, z0 not zero, then (x0k,y0k,z0k), is a nonzero solution of xp + yp = zp, which is a contradiction. If n is not divisible by an odd prime, then n = 2r (r > 2). If xn + yn = zn has a solution (x0,y0,z0), x0, y0, z0 not zero, then (x02r-2,y02r-2,z02r-2), is a nonzero solution of x4 + y4 = z4, which is also a contradiction.

Let us assume Fermat's result for n = 4 and let us concentrate on the Diophantine equation

(5)
xp + yp = zp,   p an odd prime.

The first organized attack on (5) for all p was made by Kummer in 1835. Kummer's idea was as follows: Let be a primitive nth root of 1, and let

Z[] = {a0 + a1 + ... + ar r | ai Z, r > 0}.

It is easy to see that Z[] is a subring of C containing 1, so that Z[] is an integral domain. The ring Z[] is a generalization of the ring Z and is called the ring of p-cyclotomic integers. (Cyclotomic means "circle dividing".) Kummer supposed it was possible to write every nonzero element of Z[] uniquely as a product of "prime elements". In order to understand the precise meaning of this statement, we need a few definitions.

Definition 2: Let R be a ring, x Rx(R - {0}). We say that x is irreducible if

(1) Whenever x = ab, a,b R, either a or b is a unit of R.

(2) x is not a unit of R.

In the ring Z, the units are ±1, so that if x Z, then x is irreducible if and only if x = ±p, where p is a prime.

Definition 3: Let R be a ring x,y R. We say that x and y are associates if x = y for some unit of R.

Thus, if x,y Z, then x and y are associates if and only if x = ±y. Using Definitions 2 and 3, we may state the fundamental theorem of arithmetic as follows:

Theorem 4: Let x Zx(Z-{0}), x ±1. Then:

(1) x can be written a product of irreducible elements of Z.

(2) If x = p1 ... ps = q1 ... qt are two expressions of x as a product of irreducible elements, then s = t and it is possible to renumber the pi so that pi and qi are associates (1 < i < s).

The fundamental theorem of arithmetic in the above form prompts us to formulate the following definition.

Definition 5: Let R be an integral domain. We say that R is a unique factorization domain (UFD) if the following two conditions are satisfied:

(1) If x Rx(R-{0}) is not a unit of R, then x can be written as a product of irreducible elements of R.

(2) If x Rx is not a unit of R, and if

x = 1 ... s = 1 ... t

are two expressions of x as a product of irreducible elements, then s = t and it is possible to renumber 1 ... s so that 1 and 1 are associates (1 < i < s).

Kummer assumed that Z[] is a UFD and on the basis of this assumption he proved that Fermat's Last Theorem is true. However Dirichlet pointed out to Kummer that his assumption was false. For example it was shown that Z[] is not a UFD. A sketch of the proof of this fact is given in the appendix. Kummer investigated the rings Z[] more closely and found that usually they are not UFD's. Therefore, Kummer's proof was invalid, owing to his own mistaken assumption. But this turned out to be a good fortune for mathematics.

In attempting to recoup the loss of unique factorization in Z[], Kummer was led to invent the notion of an ideal. And Kummer's deep results on the arithmetic of ideals of Z[] are the beginning of modern ring theory. In these sections we will study some elementary ideal theory and will indicate the connection between the theory of ideals and unique factorization. At the end we will return to Fermat's Last Theorem and will use the theoretical apparatus of this chapter to describe the achievements of Kummer somewhat more explicitly than can be done at this point.