Algebraic Numbers, Index and Norm

Index and Norm

This section equates the index of an ideal in a ring, i.e. the size of the quotient ring, to the index of a sublattice inside a larger lattice, and then to the norm of the element that generates the ideal in its ring. This assumes some familiarity with the concept of a lattice.

Let K be the fraction field of R. Assume the vector space Kⁿ is also a field E. In other words, E/K is a finite field extension.

Let b₁ through b_n be a basis for a free R module S inside E. Now S is a regular R lattice of E. Furthermore, assume S is closed under multiplication. Thus S is a subring of E.

Let w be an element of S, which generates a principal ideal H in S. Since H is an R submodule, it becomes a sublattice inside S. Note that H contains wb₁ through wb_n, linearly independent vectors in E. Therefore H is an n dimensional lattice, and a valid lattice for Kⁿ. In fact wb₁ through wb_n forms a basis for H, as a free R module and an R lattice of Kⁿ.

What is the index of H in S? A linear transformation carries b_i to wb_i, and carries S onto H. Build a matrix M, where the i^th row is the representation of wb_i, as coefficients on b₁ through b_n. Thus the entries of M lie in R. The determinant of M gives the index of H in S. In addition, M implements multiplication by w. Its determinant is therefore the norm of w in E/K. Therefore the norm of w is the index of the sublattice H in the lattice S.

Now look at the quotient ring S/H. Start by assuming R = Z. The elements of S are the linear combinations of b₁ through b_n with integer coefficients. These are essentially the lattice points in n space. The sublattice H is spanned by vectors v₁ through v_n, where v_i is wb_i, represented as coefficients on b₁ through b_n. We have n linearly independent vectors, with integer coefficients, living in an integer lattice, and defining a sublattice. Start with any point x in S and shift it by a multiple of v₁, then by a multiple of v₂, and so on through v_n, until x lies in the base cell. Each time x is translated by something in H, and this does not change the coset of x. Thus everything in S/H is represented by a lattice point in the base cell, i.e. the parallelatope spanned by v₁ through v_n. Conversely, suppose two points in the base cell represent the same coset of H in R. There difference, a vector strictly inside the base cell, would have to be part of the lattice, and that is a contradiction. Therefore lattice points in the base cell correspond to the elements of S/H.

Since each lattice point represents a unit of volume, the number of lattice points in the base cell equals the volume of that cell, which is the covolume of the sublattice. (You can see a more precise proof here.) This is the determinant of v₁ through v_n, which is the determinant of wb₁ through wb_n, which is the norm of w. Put this all together and the norm of w equals the cardinality of S/H. As a corollary, every principal ideal in S has a finite index.

Product of Conjugates

Remember that the norm of w is the product of its conjugates, raised to a power, according to the dimension of E over K(w). If w generates E, and if E/K is separable, then the index of wS in S is the norm of w, or the product of the conjugates of w. We don't require E/K to be a normal extension; just make sure you include all possible conjugates of w, whether they lie in E or not.

Technically the index is the absolute value of the norm. After all, the norm, or the determinant, could come out negative, and the cardinality of S/H is obviously positive. Since the conjugates lie in the complex plane, we are taking the absolute value of a finite product of complex numbers. This in turn is the product of the absolute values of the conjugates of w in the complex plane.

If E goes beyond K(w), and if E/K is separable, the norm of w is the product of the conjugates raised to the j power, where j is the dimension of E over K(w). Now consider all possible embeddings of S, or E, into the complex plane. These map K(w) into the complex plane, and then E into the complex plane. All the conjugates of w appear in the embeddings of K(w), and they appear again and again, for each embedding of E over K(w). Thus the norm of w is the product of the images of w under all the embeddings of E into the complex plane, and after taking absolute values, the index of H in S is the product of the absolute values of the images of w in the complex plane, under all possible embeddings of E.

When R is a Finite Extension of Z

This is probably more than we need, an unnecessary generalization, but let R be a finite integral extension of Z, with a basis of c₁ through c_m. Now S is a free R module with basis b₁ through b_n, but it is also a free Z module with basis b₁ through b_n times c₁ through c_m. In other words, S can be viewed as the lattice points in mn space.

As before, H is a sublattice of S. It is spanned by v₁ through v_n, where each is a linear combination of b₁ through b_n with coefficients in R. Expand these coefficients, and each v_i becomes a linear combination of b₁ through b_n times c₁ through c_m with integer coefficients. This gives n vectors in nm space. However, v₁ could be multiplied by c₁. This was included in the span of v₁, over R, but with integer coordinates, we have to consider c₁ v₁ separately. Do this across the board, and our n vectors become mn vectors, as each is multiplied by c₁ through c_m.

Suppose a linear combination of these mn vectors, using integer coefficients, equals 0. Group them together by v₁, v₂, v₃, etc. Since v₁ through v_n are independent, the coefficient on v₁ must be 0. This means the linear combination of c₁ through c_m that builds the coefficient on v₁ has to be 0. Since c₁ through c_m forms a basis for the free Z module R, all coefficients are 0, and the vectors are linearly independent after all. Thus H has the same dimension as S. In fact, the mn vectors, v₁ through v_n times c₁ through c_m, generate H.

As before, the index of H in S, as a lattice, equals the index of H in S, as an ideal. This is the norm of w in E/Q. Remember that the norm of the norm is the norm, so the index of H in S is the norm of w, as an element of R, with a second norm applied, to produce an integer. Once again every principal ideal of S has finite index.

Polynomial Extension

Let R be Z_p[t] and let K be its fraction field Z_p(t). As before, S is a lattice of Kⁿ. Let w generate a principal ideal H in S. The norm of w is the determinant of the matrix that implements multiplication by w. This is a polynomial in t with coefficients in Z_p. Take the degree of this polynomial and raise p to that power. Equivalently, take the index of this polynomial in R. either way you have a power of p, and thanks to a rather technical theorem in lattices over polynomials, this agrees with the index of H in S. Like number fields, the index of H in S, i.e. the size of S/H, equals the index of the norm of w in R.

The definition of a global field requires E/K to be separable. With this in mind, the product of the conjugates of w, raissed to a power determined by the dimension of E over K(w), equals the norm of w. We would like to relate this to the images of w under various embeddings of E, as we did with number fields. However, we need something analogous to the complex plane.

Let C be the algebraic closure of K, or, if you prefer, let C be the algebraic closure of the completion of K, using the valuation induced by R. Now C has a valuation that is consistent with R.

The sum of the valuations gives the valuation of the product, and so, the index of H in S is p raised to the sum of the valuations of the images of w in C, for all possible embeddings of E in C.

If you didn't understand this part, that's ok. The earlier result, on number fields embedded in the complex plane, is really the primary result.