Theory of Algebraic Integers
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By Terence Tao. NT , math. RA Tags: algebraic integer , class number formula , Dedekind zeta function , Dirichlet character , ideals , number field , unique factorisation by Terence Tao 18 comments. Analytic number theory is only one of many different approaches to number theory. Another important branch of the subject is algebraic number theory , which studies algebraic structures e.
With this perspective, the classical field of rationals , and the classical ring of integers , are placed inside the much larger field of algebraic numbers , and the much larger ring of algebraic integers , respectively. Recall that an algebraic number is a root of a polynomial with integer coefficients, and an algebraic integer is a root of a monic polynomial with integer coefficients; thus for instance is an algebraic integer a root of , while is merely an algebraic number a root of.
For the purposes of this post, we will adopt the concrete but somewhat artificial perspective of viewing algebraic numbers and integers as lying inside the complex numbers , thus.
From a modern algebraic perspective, it is better to think of as existing as an abstract field separate from , but which has a number of embeddings into as well as into other fields, such as the completed p-adics , no one of which should be considered favoured over any other; cf. But for the rudimentary algebraic number theory in this post, we will not need to work at this level of abstraction. In particular, we identify the algebraic integer with the complex number for any natural number.
Exercise 1 Show that the field of algebraic numbers is indeed a field , and that the ring of algebraic integers is indeed a ring , and is in fact an integral domain. Also, show that , that is to say the ordinary integers are precisely the algebraic integers that are also rational. Because of this, we will sometimes refer to elements of as rational integers. In practice, the field is too big to conveniently work with directly, having infinite dimension as a vector space over.
Thus, algebraic number theory generally restricts attention to intermediate fields between and , which are of finite dimension over ; that is to say, finite degree extensions of. Such fields are known as algebraic number fields , or number fields for short. Apart from itself, the simplest examples of such number fields are the quadratic fields , which have dimension exactly two over.miacisporal.tk
Algebraic integer - Wikipedia
Exercise 2 Show that if is a rational number that is not a perfect square, then the field generated by and either of the square roots of is a quadratic field. Conversely, show that all quadratic fields arise in this fashion. Hint: show that every element of a quadratic field is a root of a quadratic polynomial over the rationals. The ring of algebraic integers is similarly too large to conveniently work with directly, so in algebraic number theory one usually works with the rings of algebraic integers inside a given number field.
One can and does study this situation in great generality, but for the purposes of this post we shall restrict attention to a simple but illustrative special case, namely the quadratic fields with a certain type of negative discriminant. The positive discriminant case will be briefly discussed in Remark 42 below. Exercise 3 Let be a square-free natural number with or.
Show that the ring of algebraic integers in is given by. If instead is square-free with , show that the ring is instead given by.
Theory of Algebraic Integers
What happens if is not square-free, or negative? Remark 4 In the case , it may naively appear more natural to work with the ring , which is an index two subring of. However, because this ring only captures some of the algebraic integers in rather than all of them, the algebraic properties of these rings are somewhat worse than those of in particular, they generally fail to be Dedekind domains and so are not convenient to work with in algebraic number theory. We refer to fields of the form for natural square-free numbers as quadratic fields of negative discriminant , and similarly refer to as a ring of quadratic integers of negative discriminant.
Quadratic fields and quadratic integers of positive discriminant are just as important to analytic number theory as their negative discriminant counterparts, but we will restrict attention to the latter here for simplicity of discussion. Thus, for instance, when , the ring of integers in is the ring of Gaussian integers.
As these examples illustrate, the additive structure of a ring of quadratic integers is that of a two-dimensional lattice in , which is isomorphic as an additive group to. Indeed, a large part of basic algebraic number theory is devoted to treating the multiplicative theory of integers in number fields in a unified fashion, that naturally generalises the classical multiplicative theory of the rational integers. For instance, every rational integer has an absolute value , with the multiplicativity property for , and the positivity property for all.
Among other things, the absolute value detects units: if and only if is a unit in that is to say, it is multiplicatively invertible in. Similarly, in any ring of quadratic integers with negative discriminant, we can assign a norm to any quadratic integer by the formula. When working with other number fields than quadratic fields of negative discriminant, one instead defines to be the product of all the Galois conjugates of.
Thus for instance, when one has. Analogously to the rational integers, we have the multiplicativity property for and the positivity property for , and the units in are precisely the elements of norm one. Exercise 5 Establish the three claims of the previous paragraph. Conclude that the units invertible elements of consist of the four elements if , the six elements if , and the two elements if.
For the rational integers, we of course have the fundamental theorem of arithmetic , which asserts that every non-zero rational integer can be uniquely factored up to permutation and units as the product of irreducible integers, that is to say non-zero, non-unit integers that cannot be factored into the product of integers of strictly smaller norm. As it turns out, the same claim is true for a few additional rings of quadratic integers, such as the Gaussian integers and Eisenstein integers, but fails in general; for instance, in the ring , we have the famous counterexample.
For instance, in , the principal ideal turns out to uniquely factor into the product of non-principal ideals ; see Exercise We will review the basic theory of ideals in number fields focusing primarily on quadratic fields of negative discriminant below the fold. The norm forms 1 , 2 can be viewed as examples of positive definite quadratic forms over the integers, by which we mean a polynomial of the form.
One can declare two quadratic forms to be equivalent if one can transform one to the other by an invertible linear transformation , so that. For example, the quadratic forms and are equivalent, as can be seen by using the invertible linear transformation. Such equivalences correspond to the different choices of basis available when expressing a ring such as or an ideal thereof additively as a copy of. There is an important and classical invariant of a quadratic form , namely the discriminant , which will of course be familiar to most readers via the quadratic formula , which among other things tells us that a quadratic form will be positive definite precisely when its discriminant is negative.
It is not difficult particularly if one exploits the multiplicativity of the determinant of matrices to show that two equivalent quadratic forms have the same discriminant. Thus for instance any quadratic form equivalent to 1 has discriminant , while any quadratic form equivalent to 2 has discriminant. Thus we see that each ring of quadratic integers is associated with a certain negative discriminant , defined to equal when and when. Exercise 6 Geometric interpretation of discriminant Let be a quadratic form of negative discriminant , and extend it to a real form in the obvious fashion.
Show that for any , the set is an ellipse of area. It is natural to ask the converse question: if two quadratic forms have the same discriminant, are they necessarily equivalent? For certain choices of discriminant, this is the case:. Exercise 7 Show that any quadratic form of discriminant is equivalent to the form , and any quadratic form of discriminant is equivalent to. Hint: use elementary transformations to try to make as small as possible, to the point where one only has to check a finite number of cases; this argument is due to Legendre.
More generally, show that for any negative discriminant , there are only finitely many quadratic forms of that discriminant up to equivalence a result first established by Gauss. Unfortunately, for most choices of discriminant, the converse question fails; for instance, the quadratic forms and both have discriminant , but are not equivalent Exercise This particular failure of equivalence turns out to be intimately related to the failure of unique factorisation in the ring. It turns out that there is a fundamental connection between quadratic fields, equivalence classes of quadratic forms of a given discriminant, and real Dirichlet characters, thus connecting the material discussed above with the last section of the previous set of notes.
Here is a typical instance of this connection:. Proposition 8 Let be the real non-principal Dirichlet character of modulus , or more explicitly is equal to when , when , and when. We will prove this proposition later in these notes. We observe that as a special case of part i of this proposition, we recover the Fermat two-square theorem : an odd prime is expressible as the sum of two squares if and only if. As an illustration of the relevance of such connections to analytic number theory, let us now explicitly compute.
Corollary 9. This particular identity is also known as the Leibniz formula. Proof: For a large number , consider the quantity. Fourier Grenoble 40 , , Numdam MR Zbl MR Aparicio , Sobre unos sistemas de numeros enteros algebraicos de D.