Algebraic Numbers

In this post we revisit certain topics discussed in one of the earliest posts on this blog, namely, The Fundamental Theorem of Arithmetic and Unique Factorization. In that post we introduced certain “numbers” such as \mathbb{Z}[i], also referred to as the Gaussian integers, and \mathbb{Z}[\sqrt{-5}], which I currently do not know the name of, despite it being one of the most basic examples of “numbers” displaying “weird” behavior such as the failure of unique factorization.

In that post we have been quite vague, and it is the intention of this post to start taking on the same topics with a little more clarity and rigor.

We define two important concepts – algebraic numbers and finite degree field extensions of the field of rational numbers \mathbb{Q}. These two concepts are the objects of study of the branch of mathematics called algebraic number theory.

An algebraic number is a complex number which is the root of a polynomial with integer coefficients. The square root of -1, which we of course write as i, is an example of an algebraic number. It is a root of the equation

x^{2}+1=0

Numbers that are not algebraic numbers are called transcendental numbers. Examples of transcendental numbers are the constants \pi and e.

Given a field (see Rings, Fields, and Ideals) F, a field extension of F is another field K that contains F as a subset (or rather, a subfield). The degree of a field extension of F is its dimension (see More on Vector Spaces and Modules) as a vector space whose field of scalars is F.

It is known that every element of a finite degree field extension of the field of rational numbers \mathbb{Q} is an algebraic number. Hence, such a field extension is also called an algebraic number field.

An algebraic number which is the root of a monic polynomial with integer coefficients is called an algebraic integer. A monic polynomial is a polynomial where the term with the highest degree has a coefficient of 1. Hence, i is not only an algebraic number, but is also an algebraic integer, since the polynomial x^{2}+1 is monic. The algebraic integers in an algebraic number field form a ring. They are related to the elements of the algebraic number field in an analogous way to how ordinary integers are related to rational numbers.

The ring of Gaussian integers \mathbb{Z}[i] is the ring of algebraic integers of the algebraic number field \mathbb{Q}[i], which is made up of complex numbers whose real and imaginary parts are both rational numbers, while the ring \mathbb{Z}[\sqrt{-5}] is the ring of algebraic integers of the algebraic number field \mathbb{Q}[\sqrt{-5}], which is made up of complex numbers which can be written in the form a+b\sqrt{-5}, where a and b are rational numbers.

A unit is an element of the ring of algebraic integers of an algebraic number field which has a multiplicative inverse.As we have already seen in previous posts, it is important to identify the units in the ring of algebraic integers because we have to exclude them when we talk about unique factorization.

One of the things we can do with an algebraic number field is to study the factorization of its ring of algebraic integers. We have explored a little bit of this in The Fundamental Theorem of Arithmetic and Unique Factorization, and we have seen that in the ring \mathbb{Z}[\sqrt{-5}] the factorization into irreducible elements fails to be unique. For example, we may have

6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5})

The numbers 2, 3, 1+\sqrt{-5}, and 1-\sqrt{-5} are all irreducible in the ring \mathbb{Z}[\sqrt{-5}].

However, for certain rings called Dedekind domains, even if unique factorization into irreducible elements does not hold, the ideals of the ring may still be factored uniquely as a product of the prime ideals (see More on Ideals) of the ring. The ring of algebraic integers of an algebraic number field happens to be a Dedekind domain. We will discuss this factorization of ideals next.

We recall that an ideal of a ring is a subset of the ring which is closed under addition and multiplication by elements of the ring. In other words, it is a subset of the ring which is also a module with the ring itself as its ring of scalars. Perhaps the most simple kind of ideal is a principal ideal, written (a) for an element of the ring a, which consists of all products of a with all the other elements of the ring. We may also say that the ideal (a) is the set of all multiples of a.

In the ring of ordinary integers \mathbb{Z}, all ideals are principal ideals. However, this may not be true for more general rings. For example, in the ring \mathbb{Z}[\sqrt{-5}], consider the set of linear combinations of 2 and 1+\sqrt{-5}, i.e. the set of elements of \mathbb{Z}[\sqrt{-5}] which can be written as a(2)+b(1+\sqrt{-5}), where a and b are elements \mathbb{Z}[\sqrt{-5}]. This set, written (2, 1+\sqrt{-5}), forms an ideal, but this ideal is not a principal ideal. It is not the set of multiples of a single element. However, it is closed under addition and multiplication by any element of \mathbb{Z}[\sqrt{-5}].

 Given two ideals \mathfrak{a} and \mathfrak{b} in some ring, the product \mathfrak{a}\mathfrak{b} is the set of all elements of the ring which can be written as a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+... where a_{1}, a_{2}, a_{3},... are elements of the ideal \mathfrak{a} and b_{1}, b_{2}, b_{3},... are elements of the ideal \mathfrak{b}.

We can now state the following “ideal-theoretic” analogue of the fundamental theorem of arithmetic (quoted from the book Algebraic Number Theory by Jurgen Neukirch):

Every ideal of \mathcal{O} different from (0) and (1) admits a factorization

\mathfrak{a}=\mathfrak{p}_{1}...\mathfrak{p}_{r}

into nonzero prime ideals \mathfrak{p}_{i} of \mathcal{O} which is unique up to the order of the factors.

Here the symbol \mathcal{O} refers to the ring of algebraic integers of an algebraic number field.

We recall once again our example showing the failure of unique factorization in \mathbb{Z}[\sqrt{-5}]:

6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5})

If we instead consider ideals instead of individual elements, we would have

(6)=(2)(3)=(1+\sqrt{-5})(1-\sqrt{-5})

(Note: Parentheses are used to denote principal ideals in abstract algebra and algebraic number theory. However, they are also used to denote multiplication of expressions, as in basic arithmetic and algebra. Hopefully the intended purpose of the parentheses will be obvious from the context and will not cause too much confusion for the reader. In the examples above, we have first used them for individual elements of the ring, and later on, for ideals, which are sets of elements of the ring.)

But the ideals in the last expression can be factored even further:

(2)=(2,1+\sqrt{-5})(2,1-\sqrt{-5})

(3)=(3,1+\sqrt{-5})(3,1-\sqrt{-5})

(1+\sqrt{-5})=(2,1+\sqrt{-5})(3,1+\sqrt{-5})

(1-\sqrt{-5})=(2,1-\sqrt{-5})(3,1-\sqrt{-5})

Therefore, the principal ideal (6) admits a unique factorization as a product of ideals as follows:

(6)=(2,1+\sqrt{-5})(2,1-\sqrt{-5})(3,1+\sqrt{-5})(3,1-\sqrt{-5})

We turn next to the definition of the class number of an algebraic number field, which was given a passing mention in The Fundamental Theorem of Arithmetic and Unique Factorization. The class number “measures” in some way the failure of unique factorization, and if its value is equal to 1, then unique factorization holds (this also means that all ideals in the ring of algebraic integers of the algebraic number field are principal ideals).

To define the class number, we first have to introduce the concept of a fractional ideal. A fractional ideal is a module which is obtained by taking the linear combinations of products of a finite number of elements of an algebraic number field with its ring of algebraic integers. Note that these elements need not be an algebraic integer itself. For example, the set

...-\frac{3}{2}, -1, -\frac{1}{2}, 0, \frac{1}{2}, 1, \frac{3}{2}...

is obtained by taking the products of the rational number \frac{1}{2} with the ordinary integers. We write it as (\frac{1}{2}), and, in analogy with principal ideals, we refer to such fractional ideals which are “generated” by a single element as principal fractional ideals. It is a property of fractional ideals that one can multiply them by a certain algebraic integer (which is an element of the ring of algebraic integers of the algebraic number field to which it belongs) and get back the ring of algebraic integers of the algebraic number field. For the example above, we can multiply each element by 2 and get back the ordinary integers.

The fractional ideals, including the principal fractional ideals, form a group (see Groups) under multiplication. The ideal class group is then the group obtained by taking the quotient (see Modular Arithmetic and Quotient Sets) of the group of fractional ideals by the group of principal fractional ideals. The ideal class group only has a finite number of elements (called ideal classes), and this number is called the class number.

There is another way to define the ideal classes. We will say that two ideals \mathfrak{a} and \mathfrak{b} are equivalent, written \mathfrak{a}\sim \mathfrak{b}, if there exist principal ideals (a) and (b) such that (a)\mathfrak{a}=(b)\mathfrak{b}. The ideals that are equivalent to each other then form an equivalence class, and these equivalence classes are the ideal classes. The set of ideal classes form a group, which is the ideal class group.

The ring \mathbb{Z}[\sqrt{-5}], which does not possess unique factorization (of elements) has two ideal classes – the class of principal fractional ideals, and another class, which includes the ideal (2, 1+\sqrt{-5}). Hence its class number is 2.

In summary, algebraic number fields are not always uniquely factorizable into irreducible elements. The class number (which requires the concept of ideals to be properly defined) allows us to somehow “measure” the failure of unique factorization. However, despite the failure of factorization of elements, there is always the uniqueness of factorization for ideals.

This makes up the basics of the subject of algebraic number theory, which I find to be interestingly named – on one hand, it is “algebraic” number theory, which means that it uses concepts from abstract algebra to study numbers. On the other hand, it is “algebraic number” theory, which means that it is the study of algebraic numbers, which we have defined above as the numbers that are zeroes of polynomials with integer coefficients. Algebraic number theory is one of the oldest and most revered branches of mathematics, and has developed consistently and grown in beauty and elegance throughout history – including in modern times.

References:

Algebraic Number Theory in Wikipedia

Algebra by Michael Artin

A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen

Algebraic Number Theory by Jurgen Neukirch

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