Defining Integers Via Norm Equations
In this section we explain some ideas behind our definitions of integers in . The central part of our construction is a norm equation which has no solutions if a field element in question has “forbidden” poles. (In an effort to simplify terminology we transferred some function field terms to this number field setting.) The first practitioners of this method were J. Robinson using quadratic forms and R. Rumely using more general norm equations. The author of this paper has generally employed a distinct variation of the norm method. More specifically, as explained below, the bottom field in the norm equation is not fixed, but is allowed to vary depending on the elements involved. As long as the degree of all extensions involved is bounded, such a “floating” norm equation is still (effectively) translatable into a system of polynomial equations over the given field. To set up the norm equation, let
- • q be a rational prime number,
- • K be a number field containing a primitive q-th root of unity,
- • pK be a prime of K not dividing q,
- • b e K be such that ordPKb = —1,
- • c e K be such that c is integral at pK and is not a q-th power in the residue field of Pk ,
and consider bxq + bq. Note that ordPK (bxq + bq) is divisible by q if and only if ordPKx > 0. Further, if x is an integer, all the poles of bxq + bq must be poles of b and are divisible by q. Assume also that all zeros of bxq + bq and all zeros and poles of c are of orders divisible by q and c = 1 mod q3. Finally, to simplify the situation further, assume that either K has no real embeddings or q > 2. Now consider the norm equation
Since pK does not split in this extension, if x has a pole at pK, then ordPKbxq + bq = 0 mod q, and the norm equation has no solution y in K (qc). Further, if x is an integer, given our assumptions, using the Hasse Norm Principle we can show that this norm equation does have a solution. Our conditions on c insure that the extension is unramified, and our conditions on bxq + bq in the case x is an integer make sure that locally at every prime not splitting in the extension the element bxq + bq is equal to a q-th power of some element of the local field times a unit. By the Local Class Field Theory, this makes bxq + bq a norm locally at every prime.
For an arbitrary b and c = 1 mod q3 in K, we will not necessarily have all zeros of bxq + bq and all zeros and poles of c of orders divisible by q. For this reason, given x, b, c e K we consider our norm equation in a finite extension L of K and this extension L depends on x, b, c and q. We choose L so that all primes occurring as zeros of bxq + bq or as zeros or poles of c are ramified with ramification degree divisible by q. We also take care to split pK completely in L, so that in L we still have that c is not a q-th power modulo any factor of pL. This way, as we run through all b, c e K with c — 1 = 0 mod q3, we “catch” all the primes that do not divide q and occur as poles of x.
Unfortunately, we will not catch factors of q that may occur as poles in this manner, because our assumption on c forces all the factors of q to split into distinct factors in the extension. Splitting factors of q into distinct factors protects us from a situation where such primes may ramify and cause the norm equation not to have solutions even when x is an integer. Elimination of factors of q from the denominators of the divisors of the elements of the rings we define is done separately.
The end result of this construction is essentially a uniform definition of the form W3 .. .3 of the ring of Q- integers, with Q containing factors of q, across all number fields containing the q-th primitive roots of unity.
Putting aside for the moment the issue of defining the set of all elements c integral at q and equivalent to 1 mod q3, and the related issue of defining integrality at factors of q in general, we now make the transition to an infinite q-bounded extension Kinf by noting the following. Let K c Kinf, let pK be a prime of K such that pK does not divide q, let x e K and let ordPKx < 0. Since by assumption pK is q- bounded, it lies along a path in its factor tree within Kinf, where the order at q of local degrees eventually stabilizes. To simplify the situation once again, we can assume that it stabilizes immediately past K. So let N be another number field with K c N c Kinf. In this case for some prime pN above pK in N, we have that ordqe(pN /pK) = ordqf (pN/pK) = 0.(Here e(pN /pK) is the ramification degree and f (pN/pK) is the relative degree.) Now, let b, c e K be as above and observe that c is not a q-th power in the residue field of pN while ordPN (bxq + bq) ф 0 mod q. Thus the corresponding norm equation with K replaced by N and eventually by Kinf in (3.10) has no solution. Of course when x is an integer and we have a solution to our norm equation in K, we also have a solution in Kinf.
Note that for each prime pK of K, at every higher level of the tree we need just one factor with the local degree not divisible by q to make the norm equation unsolvable when pK appears in the denominator of the divisor of x. Hence having one q-bounded path per every prime of K is enough to make sure that no prime of K not dividing q occurs as a pole of any element of K in our set.
Unfortunately, if we go to an extension of K inside Kinf, some primes of K will split into distinct factors and can occur independently in the denominators of the divisors of elements of extensions of K. Thus, in the extensions of K inside Kinf we have to block each factor separately. This is where the “hereditary” part comes in. We need to require the same condition of q-boundedness for every descendant in the factor tree of every prime of K not dividing q, insuring integrality at all factors of all K-primes not dividing q.
The main reason that only one q-bounded path per prime not dividing q is enough to construct a definition of integers, is that the failure of the norm equation to have a solution locally at any one prime is enough for the equation not to have solutions globally. Conversely, in order to have solutions globally, we need to be able to solve the norm equations locally at all primes. As already mentioned above, the reason we require c to be integral at q and equivalent to 1 mod q3 is to make sure that factors of q do not ramify when we take the q-th root of c. Just making c have order divisible by q at all primes does not in general guarantee that factors of q do not ramify in such an extension. If any factor of q does ramify, then not all local units at this factor are norms in the extension, and making sure that the right side of the norm equation has order divisible by q at all primes might not be enough to guarantee a global solution. Hence we need to control the order of c — 1 at all factors of q at every level of the factor tree simultaneously, necessitating a stronger assumption on q, than on other primes.
Depending on the field we might have a couple of options as far as integrality at q goes. If q happens to be completely p-bounded in our infinite extension for some p = q, then we can pretty much use the same method as above with p-th root replacing the q-th root. The only difference is that, assuming we have the primitive p-th root of unity in the field, by definition of a complete p-boundedness, we can fix an element c of the field such that c is not a p-th power modulo any factor of q in any finite subextension of Kinf containing some fixed number field. We can also fix an element b of the field such that the order of b at any factor of q is not divisible by p in any finite subextension of Kinf containing the same fixed number field as above. Using such elements c and b we can get an existential definition of a subset of the field containing all elements with the order at any factor of q bounded from below by a bound depending on b and p. If ramification degrees of factors of q are altogether bounded, then we can arrange for this set to be the set of all field elements integral at factors of q, but in a general case the bound from below will be negative. In this case to obtain the definition of integrality we will need one more step.
Before going back to infinite extensions, we would like to make a brief remark about the sets definable by our methods over number fields. First of all, over any number field all primes are completely p-bounded for every p, and the ramification degree of factors of q is altogether bounded. So we can produce an existential and uniform (with parameters) definition of integrality at all factors of q. Note also that the complement of such a set is also uniformly existentially definable with parameters using the same method. So, in summary, we now obtain a uniform definition of the form W3... 3 of the ring of integers of any number field with a q-th primitive root of unity. This result is along the lines of B. Poonen’s result in , though his method is slightly different from ours since it uses ramified primes rather than non-splitting primes to obtain integrality formulas and restricts the discussion to q = 2 and quadratic forms. As B. Poonen, we can also use q = 2 and thus have a two-universal quantifier formula uniformly covering all number fields, but in this case if K has real embeddings, we need to make sure that c satisfies some additional conditions in order for the norm equations to have solutions.
Returning now to the case of infinite extensions, we note that, assuming q is p- bounded we now have a uniform first-order definition with parameters of algebraic integers across all q-bounded algebraic extensions of Q where q is completely p- bounded. However, for the infinite case we may require more universal quantifiers. The number of these universal quantifiers will depend on the whether the ramification degree of factors of q is bounded and on whether q has a finite number of factors.
The only case left to consider now is the case where q is not completely p- bounded for any p = q but is completely q-bounded. This case requires a somewhat more technically complicated definition than the case where we had a requisite p. In particular, we still need a cyclic extension (once again of degree q), where all the factors of q will not split. Such an extension does exist, but we might have to extend our field to be in a position to take advantage of it.