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Mathematics

Algebraic Number Theory and the Ring of Integers

Quick fact

In the ring of integers of the number field ℚ(√-5), the number 6 has two distinct factorizations into irreducibles, showing that the Fundamental Theorem of Arithmetic fails in this ring of algebraic integers.

Why this is interesting

You know that 6 can be factored as 2 × 3, and that this is the only way. But what if we extend our number system to include √-5? Suddenly, 6 = 2 × 3 = (1+√-5)(1-√-5). Which factorization is the 'real' one?