Fundamental Theorem of Arithmetic - Generalizations

Generalizations

The first generalization of the theorem is found in Gauss's second monograph (1832) on biquadratic reciprocity. This paper introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers. It is now denoted by He showed that this ring has the four units ±1 and ±i, that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes.

Similarly, in 1844 while working on cubic reciprocity, Eisenstein introduced the ring, where is a cube root of unity. This is the ring of Eisenstein integers, and he proved it has the six units and that it has unique factorization.

However, it was also discovered that unique factorization does not always hold. An example is given by . In this ring one has

Examples like this caused the notion of "prime" to be modified. In it can be proven that if any of the factors above can be represented as a product, e.g. 2 = ab, then one of a or b must be a unit. This is the traditional definition of "prime". It can also be proven that none of these factors obeys Euclid's lemma; e.g. 2 divides neither (1 + √−5) nor (1 − √−5) even though it divides their product 6. In algebraic number theory 2 is called irreducible (only divisible by itself or a unit) but not prime (if it divides a product it must divide one of the factors). Using these definitions it can be proven that in any ring a prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers every irreducible is prime". This is also true in and but not in

The rings where every irreducible is prime are called unique factorization domains. As the name indicates, the fundamental theorem of arithmetic is true in them. Important examples are Euclidean domains and principal ideal domains.

In 1843 Kummer introduced the concept of ideal number, which was developed further by Dedekind (1876) into the modern theory of ideals, special subsets of rings. Multiplication is defined for ideals, and the rings in which they have unique factorization are called Dedekind domains.