Dedekind Domains, The Index of an Ideal
The Index of an Ideal
Let the index of an ideal H in a dedekind domain R be the cardinality of R/H.
For consistency let the index of the 0 ideal be 0.
This is quite nice in Z, where the index is the absolute value of the generator.
If H contains J, represent the elements of R/J
as cosets of J in H cross cosets of H in R.
Thus the index of J is the index of H times the index of J in H.
Index and Product
The index of a product of ideals is the product of the indices.
We will demonstrate this for H*M, where M is a maximal ideal.
If this holds then the index of GH is the product of the indexes of every maximal ideal (including multiplicities) that contributes to GH.
The same number can be realized by considering the prime factors of G in aggregate,
then the prime factors of H.
Thus the index of GH equals the index of G times the index of H.
Let R/M be the field K.
Tensor H with R/M, as R modules, giving H/MH, which is a K vector space.
Since H cannot equal MH, the dimension is positive.
If the dimension is greater than 1,
let x generate a one dimensional subspace inside the larger K vector space.
Remember that x ultimately comes from H.
let W be the ideal spanned by MH and x.
Now W is an ideal properly between H and MH, which is impossible.
Therefore H/MH has dimension 1, and is isomorphic to K.
The index of MH is |K| times the index of H,
where |K| is the index of M.
That completes the proof.
When each R/M has a finite index,
each ideal has a finite index.
Fractional Ideals
If an ideal H has index n, let the inverse of H have index 1/n.
Do this for all ideals,
and build a group homomorphism from the nonzero ideals into the positive rational numbers.
A fractional index has no meaning, interms of cosets or cardinality,
but it is often convenient.