Integral Rings, Finite Index Propagates Upward

Finite Index Propagates Upward

If R is dedekind, and all nonzero ideals in R have finite index, the same holds in S. Such is the case when R = Z. Remember that if the index of each prime ideal in R is finite, then the index of each ideal is finite.

Let b₁ through b_n be a basis for E/F that is contained in S. Let u lie in a nonzero ideal H in S. Divide each basis element by u, and some multiple of this element lies in S. In other words, v_ib_i/u lies in S. Thus v_ib_i lies in H.

Remember that some d can be chosen, so that the basis, divided by d, spans all of S. Divide each b_i by d, and multiply each v_i by d.

A free R module, spanned by v₁b₁ through v_nb_n, lives inside a free R module spanned by b₁ through b_n. The quotient module is isomorphic to the direct product of R/v₁ through R/v_n. All ideals in R have finite index, so the quotient module is finite.

The ring S, and the ideal H, map to submodules of this quotient module, therefore the cosets of H in S, or the index of H in S, is finite.

Finite Ideal Count Propagates Upward

Suppose there are finitely many primes in R with index < c for any positive integer c. The same is true for all the ideals of R.

Let H be an ideal in R with index < c. Write H as a product of prime ideals, and note that each prime ideal has index at least 2. There are at most log₂(c) primes in the factorization. Each prime in the factorization has index < c, and there are finitely many of these to choose from. We are choosing at most log₂(c) items from a finite set, allowing for duplicates, and there are finitely many ways to do this. Therefore the number of ideals in R with index < c is finite.

We would like to propagate this result up to S. As shown above, we only need prove the result for prime ideals, whence it applies to all ideals in S. This assumes S is dedekind, but that follows from R being dedekind, as discussed earlier.

Let Q be a prime over P, where the index of Q is less than c. Since R/P embeds in S/Q, the index of P is less than c. There are finitely many choices for P, and each such P has finitely many primes Q lying over it. (We'll prove this in the next section.) Therefore the primes of S with index < c is finite, and the same holds for the ideals of S.

The Valuation Propagates Upward

Let R be a dvr, whence S is a pid. We already know the valuation on F can be extended to E, but this is only an existence proof - not very helpful. In this case we can be more specific.

Let P be the prime in R that seeds the valuation, and let Q be a prime in S over P. Localize about Q and build a valuation ring inside E. In fact S_Q is a dvr. Now Q, the prime ideal or its generator, seeds a valuation that is consistent with P. Write v(Q) = v(P)/n for some integer n. (This n need not agree with the dimension of E/F.) The extended valuation on E is discrete, like that of F, but it has a finer granularity, like drawing the half, quarter, and eighth inch lines on a ruler.