Fundamental Theorem of Arithmetic

It’s hard to understand anything from just one example. One of the reason for studying other planets is that it helps us understand Earth. It can even be helpful to have more examples when the examples are purely speculative, such as xenobiology, or even known to be false, i.e. counterfactuals, though here be dragons.

The fundamental theorem of arithmetic seems trivial until you see examples of similar contexts where it isn’t true. The theorem says that integers have a unique factorization into primes, up to the order of the factors. For example, 12 = 2² × 3. You could re-order the right hand side as 3 × 2², but you can’t change the list of prime factors and their exponents.

I was unimpressed when I first heard of fundamental theorem of arithmetic. It seemed obvious, and hardly worth distinguishing as the fundamental theorem of arithmetic. (If not this, what would the fundamental theorem of arithmetic be? Any candidates? I didn’t have one, and still don’t.)

In order to appreciate the unique factorization property of the integers, mathematicians created algebraic structures analogous to the integers, i.e. rings, in which unique factorization may or may not hold. In the language of ring theory, the integers are a “unique factorization domain” but not all rings are.

An easy way to create new rings is to extend the integers by adding new elements, analogous to the way the complex numbers are created by adding i to the reals.

If you extend the integers by appending √-5, i.e. using the set of all numbers of the form a + b√-5 where a and b are integers, you get a ring that is not a unique factorization domain. In this ring, 6 can be factored as 2 × 3 but also as (1 + √-5)(1 – √-5).

However, you can extend the integers by other elements and you may get a unique factorization domain. For example, the Eisenstein [1] integers append (-1 + i√ 3)/2 to the ordinary integers, and the result is a unique factorization domain.

More fundamental theorem posts

[1] That’s Eisenstein, not Einstein. Gotthold Eisenstein was a 19th century mathematician who worked in analysis and number theory.

One thought on “Fundamental Theorem of Arithmetic

  1. I suppose the Euclidean division theorem would be a good candidate – it implies unique factorization and it is more specific to the integers.

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