Discriminant of a number field is arguably its most important numerical invariant. Quite closely connected to it is the different ideal. Here we discuss the most basic properties of these two concepts. The discussion mostly follows this expository paper by K. Conrad, but also takes from a series of exercises in chapter 3 of Number Fields.
Let be a number field of degree over . By standard results of field theory there are precisely embeddings of into , call them . We recall a standard definition:
Definition: For any elements we define the discriminant of these elements to be the square of the determinant of . We denote it by .
It’s easy to see doesn’t depend on the order of nor the order of . Also, , where denotes the trace. From there it’s straightforward that the discriminant lies in , and we can also deduce that iff are linearly independent over . Lastly, if are elements which are -linear combinations of represented by a matrix , then we easily see . In particular, if and are two bases of the same additive group, then they have the same discriminant. Therefore, it makes sense to speak of the discriminant of an additive subgroup to be the discriminant of any of its bases.
The most important additive subgroup of a number field is its ring of integers . The discriminant of this ring will also be sometimes called the discriminant of the field and denoted by .
Dual basis and codifferent
Consider an additive subgroup generated by a basis of . Then the matrix is invertible (since its determinant is nonzero discriminant). Considering the columns of its inverse as coefficients of a linear combination of , so constructed elements, call them , satisfy . Moreover, by uniqueness of matrix inverse, these elements are defined uniquely. We verify that the are linearly independent, hence form a basis: if , then , so the linear combination is trivial.
Definition: Given a basis , we call the basis its dual basis. We call the additive group generated by them the dual group and is denoted by .
Note it’s not immediately clear that this definition is independent of the basis of we choose. The first result which we properly state and prove will imply this.
Proposition 1: is precisely the set of such that .
Proof: Let . Since dual basis is a basis, we can write , and iff . At the same time, . It clearly follows that if , then . Conversely, if , then for any we have , so .
It is possible to explicitly give the dual basis if the basis is of the form with , i.e. its minimal polynomial over has integer coefficients.
Proposition 2: Let be the minimal polynomial of . Then is the dual basis of . Moreover, .
Proof: Let be the conjugates of in . It’s easy to see is a monic polynomial in of degree , and if we divided by , the coefficients would be . For consider the polynomial
It’s easy to see that each term is for and for . Hence this polynomial of degree smaller than agrees with polynomial at places, so the polynomials must be equal. Comparing coefficient of we get
but the left hand side is precisely , showing the first claim. To see the second claim, recall that are monic polynomials of degree , so we can show by induction that . We omit the details.
The construction of dual additive group also preserves the property of being a fractional ideal. More precisely:
Proposition 3: Let be a fractional ideal. Then (considered as the dual of the additive group) is also a fractional ideal. Moreover, . [recall that is defined as the set of these for which . In this post we establish that ]
Proof: Fix any any . For we have . But, since is a fractional ideal, , so , so . This shows is a fractiona ideal.
For the second part, suppose first . For any we have , so , so . Hence, . Hence . For the converse, pretty much this argument in reverse works.
Previous proposition shows that duals work a bit like inverses. By taking duals inverse, we get another important ideal.
Definition: Let be a fractional ideal. We define the different of to be . In particular, we call the different of the different of .
Note that , so is an ideal in . From proposition 3 we immediately have , hence for the most part we only have to focus our attention of . It takes particularly simple form when – by proposition 2 we then have , being the minimal polynomial of .
Recall the definition of the norm of an ideal: .
Proof: First we note that for fractional ideals and we have an isomorphism of rings (this is quite straightforward to establish). In particular, taking this gives . In particular, . It is well-known that for two free abelian groups of the same rank, is the absolute value of the determinant of a transformation taking basis of to the basis of . In our case, take an integral basis of and its dual basis. We write . Then . In other words, the transformation matrix is precisely the matrix , whose determinant is . This establishes the theorem.
The different is important when working with ramification of primes in a number field. As will be proven in the future post, different ideal is divisible precisely by prime ideals which which are ramified in . In the next blog post we shall establish, among other things, this result in normal extensions, together with precise formula for the exponent of this prime.
As a closing remark, it is worth poining out that the whole theory of discriminants and differents can be built in extensions for different from the field of rational numbers, although things get a lot more technical, since, for example, discriminant has to be considered as an ideal and not a single element. I hope to one day cover the theory of general discriminants and different ideals in another blog post or two.