The basic object in number theory is perhaps the algebraic number field and the corresponding algebraic integer ring.
First things first, what is an algebraic number over the ? Or more generally, over a certain field ? Algebraic means polynomials, so this algebraic number must satisfy a polynomial with coefficients in . To define an algebraic integer, we have to know what is the algebraic ring of . It is not so obvious, and perhaps we should define what is the algebraic integer ring for the most basic field, the rational number field. In fact, it is first we know that there is the integer numbers, then we can define what is the algebraic integer ring over a number field.
By a number field, we mean a field which contains the rational number field. By an algebraic number field, we mean an finite algebraic extension of the rational number field. Suppose that two algebraic number fields, we define the degree of the extension of over to be the dimension of viewed as a vector space over , . In this post, we always assume that the degree of extension to be finite.
Note that all algebraic number fields are contained in the complex number field . This is an important observation, since later we will talk maps from a field to another that we don’t know exactly, and in this case, we always assume that this field is mapped to the complex number field, although we can show that in fact this kind of maps can’t get to every element in .
Having defined algebraic number fields, then the natural things to do are of course the homomorphisms. Note that a homomorphism between two fields is very special, it is either injective, either zero. This looks like the morphisms between simple groups, simples modules, simple algebras, etc.. What should we first consider as homomorphisms? One such between two given fields? Perhaps it is a good idea. But it is better to first consider only homomorphisms starting from a given field to the complex number field.
This motivates the following definition: suppose that is an algebraic number field, then to be all the homomorphisms of fields from to . Note that, if , then it send to , and thus sends the rational numbers to itself. So this gives that . Noting that is a finite extension of , so this motivates again the following consideration,
where are two algebraic number fields.
How can we determine a homomorphism of fields? First of all we can view this homomorphism as a homomorphism of -vector spaces. So easily we get that a such homomorphism is determined by the action of this morphism on a basis of vector spaces for . So this inspires us to find what a basis of a field looks like. First, we consider the field . An obvious basis for this -vector space is . And if is a homomorphism from to the complex number field, then it is determined by . This is a very general idea, that is, if is algebraic over a field , then for any , it is determined by . So there is a one-to-one correspondence between and the set of possible values of . So we must find this set of possible values. Note that if a polynomial annulate , then we must have that . So, the possible values of are those satisfying the minimal polynomial for . So, there are not many possible values, and we get that(if we set the minimal polynomial of over to be ):
A little digression before going on. An efficient way of viewing the field extension is perhaps the point of view of algebras. If are two fields, then is in fact an algebra over (which is, moreover, divisible and commutative). And suppose further that , is a field, then in the vector space point of view, we can choose a -basis for () and a -basis for (). So how do view as a -vector space? In fact this is trivial since is contained in . Then what is the -basis for ? The answer is .
Return to our morphism stuff. In this way, we can show that
This result is not hard to prove. We can show it by using recurrence on .
An inverse problem is that, given a element , can we extend it to an element of ? This is not difficult, either. We can extend to () in using the same method as above. So, we have that,
For any , we can extend it to .
Now let’s have a closer look at . Its elements fix . The question is that can these morphisms determine ? Suppose that , then according to the above argument, we can find a morphism such that (the roots of the minimal polynomial of are all distinct, which is due to the fact that the characteristic of these number fields is ). Then we can extend this to , thus in this way, we have shown that
Sometimes we also denote . Using the same idea, we can get the primitive element theorem:
If are two fields, then there exists such that .
We can extend to , and we consider the set . An element is in this set means that it is not fixed by any . If , then we can extend to to get some morphism , but this time , a contradiction. It remains to show that is not empty. Viewed as -vector spaces, is just the complement of an union of a finite number of hyperplanes, which is not empty since is infinite. Thus we proved the primitive element theorem.
There is a particular type of morphism in from to itself. Suppose that , then the morphism . This morphism is -linear, that is , so for any subfield , is also -linear. This is one of the most important morphisms over a field. From this morphism we can define many important maps on .
Suppose that two algebraic number fields, and . Then is a -vector space, and thus is in fact a -transformation on , that is . So we can define the trace, the determinant, the characteristic polynomial of , as follows,
For , we define , , and .
Note that the norm of , is just the determinant of . What is more, .
The idea of defining these quantities is to control in using the informations from . It is obvious that is additive, while is multiplicative. What is more, is exactly the minimal polynomial of over , and thus, we have that . One last point, for , there are exactly many elements in , each corresponds to an conjugate of . According to their definition, these conjugates of must satisfy the same minimal polynomial as , so counting the cardinality, we have that . Using the same idea, we have that , and .
There is another quantity concerning the whole over . Suppose are elements, then we define the discriminant of these elements to be the discriminant of the matrix whose entries are . That is, . The motivation for this quantity is that, if these elements for a basis of over , then any number can be written as . Now we want to know what these coefficients are. What do we usually do in an Euclidean space? One natural way is to consider make scalar product of both sides with this basis. But, here we haven’t defined any scalar product. But recall what is a scalar product. A scalar product is, first of all, a bilinear form on the vector space to the ground field. Note we have seen something linear, yes, that is the trace function on , at that time we say that it is additive, which is, in some sense equivalent to linearity. How do we define a bilinear form with the help of the trace function. Note that is already bilinear, so we can take the trace of the image, thus we can define . So for the basis, we naturally have a matrix . So to find these , we have to consider the linear system . Is this system invertible? Clearly this depends on the invertibility of the basis matrix . How to determine the invertibility of this matrix? A little remark perhaps helps. Note that . If we numerate these as (), then we see that this matrix is in fact the product of two matrices, or more precisely the product of a matrix with its transpose. This matrix is , a row of looks like , and . So, if these are -linearly independent, then we must have that , which implies that is invertible, so . The converse is also true. Another character of discriminant is the relation with the discriminant of a polynomial. If , then for the minimal polynomial of , , its discriminant is just where these are just the roots of this polynomial, and thus the conjugates of . So if , then we must have that which is Vandermonde matrix, thus it is clear that .
Next we want to say something about the integer ring of , an algebraic number field. We set to be the set of algebraic integers (over ). And we define . This is the so-called algebraic integer ring of . Then the most important problem is how to decide if an element in lies in . One tends to believe that there is a simple description of , for example for some . This indeeds works for the quadratic fields over , just like the post ‘quadratic integer ring’, where all algebraic integer ring over a quadratic field looks like . But in general, this is not the case, as we shall see an example.
We have said that the quantities like trace, norm, determinant gives some information for an element in is using the information only in . Now for an algebraic integer, can we say something about it using only the information from ? Note that the trace and the norm use only partial information of the characteristic polynomial, so perhaps a good candidate for this task is the characteristic polynomial or minimal polynomial(in , they coincide). Note that, if is an algebraic integer, then all its conjugates are algebraic integers, too. So, its minimal polynomial is just . Noting that an algebraic integer is a rational number, then it must be an integer. So, we have that . Conversely, we know that the characteristic polynomial of in over annulates , so if this polynomial has integer coefficients, then must be an algebraic integer. If , then note that the characteristic polynomial is just a power of the minimal polynomial, thus we get that
For an element , we have that if and only if , if and only if .
Now let’s say something more about this algebraic integer ring . According to the primitive element theorem, we can have that . Since satisfies some polynomial, , then multiplying both sides by , we see that is in fact an algebraic integer of , but still we have that . So we can suppose directly that . So, we must have that . Can we control this algebraic integer ring from superior? The answer is yes. In deed, for any , we can write . We must show that the denominators of these for any algebraic integer in can’t be too large. This is not easy to see. But perhaps we should first get these out of this expression. How? Very simple, using the matrix . Note that, the entries of have a largest denominator, say . Then the vector has integer components since are all algebraic integers. Thus we get that the denominators of these for any can not exceed , thus showing that . Now that is sandwiched between two lattices of , thus it must has a -basis with elements. That is to say, .
But the problems is that in general we can not write this as for some element . Here is such an example. We take where two distinct square-free integer such that . We set . We first show that . Note that, since , which shows that . And since are -independent, so they form a basis for over . Note any , it is determined by . But the only other conjugate of is its inverse , so we have that there are only four elements in . And we can show that . It is clear that . So for an element , we have that , that is . Similarly, we have that . So we can rewrite . In using the trace matrix corresponding to the basis , we have that these coefficients only have possible denominators like , but are odd integers, so we must have that these coefficients must also be integers, in this way we proved that . Note that , an even integer, the same for , this means that the quotient ring is isomorphic to the ring . So if , then we must have that . But consider the ring homomorphism from to . Counting the cardinality, we can see that this is impossible. So .