prime ideal factors

[AntonioLao]

After 1843 Kummer (1810-93) had already made logical definitions of integer, prime, and divisor. However, he made the wrong assumption that unique factorization exists for these classes of algebraic numbers he had introduced. He made it clear in a preprint he sent to a contemporary mathematician by the name of Dirichlet (1805-59) who was a student of Gauss and Jacobi that unique factorization is necessary to prove Fermat’s last theorem (1601-65). By this time it was already almost 200 years old mystery which was finally proved by Wiles in 1995, 330 years later. Dirichlet told Kummer that unique factorization holds only for certain primes. Recognizing the soundness of Dirichlet criticism, Kummer invented a theory of ideal numbers in 1844.

The whole number 6 is the product of two prime factors: 2 and 3. In the theory of ideal numbers, 6 is also the product of two ideal conjugates (1+√5i) and (1-√5i) or (2+√2i) and (2-√2i). On the other hand, 2 is the product of (1+i) and (1-i) and 3 is that of (1+√2i) and (1-√2i). Therefore 6 has prime ideal factors of (1+√5i), (1-√5i) or (2+√2i), (2-√2i) or (1+i), (1-i), (1+√2i), and (1-√2i).

This inductive process clearly demonstrated that the real numbers including all primes are the products of ideal factors and some can have different phase groups of prime ideal numbers.

[mkirkpatrick]

Numbers are a odd thing?Is not six just just one repeated?



regards michael.

[AntonioLao]

Is not six just just one repeated?

It belongs to the one group of integers but there can be more than one group of symmetry.