The next few posts will be me detailing some interesting results in the area of Maths that I hope to specialise in: Algebraic Number Theory. The first result will be the unique prime factorisation of ideals. But first, what is an ideal?

## Ideals

If you’re not familiar with the definition of a **ring** click here, as we’ll need this for the following discussion.

An ideal, *I*, is a subset of a ring *R *if :

- It is closed under addition and has additive inverses (i.e. it is an additive subgroup of (
*R*, +. 0);
- If
*a* ∈ *I* and *b* ∈ *R*, then *a* · *b* ∈ *I*.

*I* is a proper ideal if *I *does not equal *R*.

## Ring of Integers

In order to prove the result, I need to introduce the concept of number fields and the ring of integers. A *number field* is a finite field extension over the rationals.

**Field Extension: **a field extension *F* of a field *E* (written F/E) is such that the operations of *E* are those of *F *restricted to *E, *i.e. *E *is a subfield of *F*.

Given such a field extension, *F* is a vector space over *E, *and the dimension of this vector space is called the degree of the extension. If this degree is finite, we have a finite field extension.

So if *F *is a number field, *E* would be the rationals.

Suppose *F *is a number field. An **algebraic integer **is simply a fancy name for an element *a *of F such that there exists a polynomial *f* with integer coefficients, which is monic (the coefficient of the largest power of *x* is 1), such that *f*(*a*) = 0.

The algebraic integers in a number field *F* form a ring, called the ring of integers, which we denote as *O*_{F}. It turns out that the ring of integers is very important in the study of number fields.

## Prime Ideals

If *P* is a **prime ideal** in a ring *R, *then for all *x, y* in *R*, if *xy* ∈ *P,* then x ∈ *P* or y ∈ *P*. As *O*_{F} is a ring we can consider prime ideals in *O*_{F}.

## Division = Containment

We want to try and deal with ideals in the same way we deal with numbers, as ideals are easier to deal with (ideals are a sort of abstraction of the concept of numbers). After formalising what it means to be an ideal and proving certain properties of ideals, we can prove that given two ideals *I *and* J, I *dividing *J *(written *I*|*J*) is equivalent to *J* containing *I*.

## Three Key Results

Now, there are three results that we will need in order to prove the prime factorisation of ideals that I will simply state:

- All prime ideals
*P* in *O*_{F }are maximal (in other words, there are no other ideals contained between *P* and *R*). Furthermore, the converse also holds: all maximal ideals in *O*_{F }are prime.
- Analogously to numbers (elements of a number field
*F*), if *I*, *J* are ideals in *O*_{F }with *J|I, *there exists an ideal *K *contained in *I*, such that *I = JK*.
- For prime ideal
*P**,* and ideals *I,J *of *O*_{F} , *P* | *IJ* implies *P* | *I* or *P* | *J*.

## Main Theorem

**Theorem:** Let *I* be a non-zero ideal in *O*_{F }. Then *I* can be written uniquely as a product of prime ideals.

**Proof:** There are two things we have to prove: the existence of such a factorisation and then its uniqueness.

Existence: If *I *is prime then we are done, so suppose it isn’t. Then it is not maximal (by 1) so there is some ideal *J*, properly contained in *I*. So *J|I, *so (by 2) there is an ideal *K, *contained in *I*, such that *I = JK*. We can continue factoring this way and this must stop eventually (for the curious, I make the technical note that this must stop as we have an infinite chain of strictly ascending ideals, and *O*_{F }is Noetherian).

Uniqueness: If *P*_{1} · · · P_{r} = Q_{1} · · · Q_{s}, with *P*_{i} , Q_{j} prime, then we know *P*_{1} | *Q*_{1} · · · Q_{s}, which implies *P*_{1} | Q_{i} for some *i* (by 3), and without loss of generality *i = 1*. So *Q*_{1} is contained in* P*_{1}. But *Q*_{1} is prime and hence maximal (by 1). So *P*_{1} = Q_{1}. Simplifying we get *P*_{2} · · · P_{r} = Q_{2} · · · Q_{s}. Repeating this we get *r = s* and* P*_{i} = Q_{i} for all *i* (after renumbering if necessary).

## Why is this important?

For numbers, we *only* get unique prime factorisation in what is called a unique factorisation domain (UFD). Examples of UFDs are the complex numbers and the rationals. However, the integers mod 10 no longer form a UFD because, for example, 2*2 = 4 = 7*2 (mod 10).

However, we have the unique prime factorisation of ideals in the ring of algebraic integers of *any* number field. This means that we can prove many cool results by using this unique prime factorisation, which we can then translate into results about numbers in that number field. I will detail some of these in future blog posts.

M x