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Unique Factorization and Fermat's Last TheoremIn 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)
x^{2} + y^{2} = z^{2}
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(a^{2}  b^{2}), y = 2abc, z = c(a^{2} + b^{2}) 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 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
x^{n} + y^{n} = z^{n}
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
Proposition 1: It suffices to prove Fermat's last theorem for Proof: Let us assume Fermat's theorem for Let us assume Fermat's result for
x^{p} + y^{p} = z^{p}, 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
It is easy to see that
Definition 2: Let R be a ring, (1) Whenever (2) x is not a unit of R. In the ring Z, the units are ±1, so that if
Definition 3: Let R be a ring
Theorem 4: Let (1) x can be written a product of irreducible elements of Z. (2) If 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 (2) If
x = _{1} ... _{s} = _{1} ... _{t}
are two expressions of x as a product of irreducible elements, then Kummer assumed that In attempting to recoup the loss of unique factorization in 
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