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?
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 OF. It turns out that the ring of integers is very important in the study of number fields.
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 OF is a ring we can consider prime ideals in OF.
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 OF are maximal (in other words, there are no other ideals contained between P and R). Furthermore, the converse also holds: all maximal ideals in OF are prime.
- Analogously to numbers (elements of a number field F), if I, J are ideals in OF with J|I, there exists an ideal K contained in I, such that I = JK.
- For prime ideal P, and ideals I,J of OF , P | IJ implies P | I or P | J.
Theorem: Let I be a non-zero ideal in OF . 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 OF is Noetherian).
Uniqueness: If P1 · · · Pr = Q1 · · · Qs, with Pi , Qj prime, then we know P1 | Q1 · · · Qs, which implies P1 | Qi for some i (by 3), and without loss of generality i = 1. So Q1 is contained in P1. But Q1 is prime and hence maximal (by 1). So P1 = Q1. Simplifying we get P2 · · · Pr = Q2 · · · Qs. Repeating this we get r = s and Pi = Qi 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.