BOUNDED REALIZATION OF l-GROUPS OVER GLOBAL FIELDS

We use the method of Scholz and Reichardt and a transfer principle from finite fields to pseudo finite fields in order to prove the following result. Theorem: Let G be a group of order l, where l is a prime number. Let K0 be either a finite field with |K0| > l or a pseudo finite field. Suppose that l 6= char(K0) and that K0 does not contain the root of unity ζl of order l. Let K = K0(t), with t transcendental over K0. Then K has a Galois extension L with the following properties: (a) G(L/K) ∼= G; (b) L/K0 is a regular extension; (c) genus(L) < 12nl ; (d) K0[t] has exactly n prime ideals which ramify in L; the degree of each of them is [K0(ζln) : K0]; (e) (t)∞ totally decomposes in L; (f) L = K(x), with irr(x,K) = X l n + a1(t)X l −1 + · · · + aln(t), 0 < deg(a1(t)) ≤ 12nl 2n and deg(ai(t)) < deg(a1(t)) for i = 1, . . . , l. Nagoya Mathematical Journal 150 (1998), 13–62 * This research was supported by The Israel Science Foundation administered by The Israel Academy of Sciences and Humanities. Introduction Scholz [Sch] proved that if l is an odd prime, then each l-group occurs as a Galois group over Q. Here G is an l-group if the order of G is a power of l. Independently, Reichardt [Rei] gave a simpler and shorter proof to the same theorem. One can find a modern presentation of Reichardt’s proof in Serre’s course on Galois theory [Se1, §2.1]. The reason why the method of Scholz and Reichardt does not work for l = 2 is that the primitive root of unity of order 2, namely −1, belongs to Q. The same reason forced Rzedowski-Calderón and Villa-Salvador [RCV] to exclude all primes l with ζl ∈ Fq, when they proved that each l-group occurs as a Galois group over Fq(t). Here q is a power of a prime p 6= l and ζl is a primitive root of unity of order l. Shafarevich [Sh1] has overcome this difficulty. He used refined combinatorial arguments to prove that for an arbitrary prime number l, for each number field K, and for each l-group G, there exists a Galois extension L of K such that G(L/K) ∼= G. In a later work [Sh2], Shafarevich pointed out how to correct an incomplete group theoretic argument in his earlier work for the case l = 2. However, Shafarevich had to pay a price for his generalization. The combinatorial arguments forced him to allow an exponentially growing number of primes of K which may ramify in L. In contrast, as Serre [Se1, p. 9] emphasizes, the method of Scholz and Reichardt gives for a group G of order l, with l odd, a Galois extension L of Q in which only n primes ramify. Although Rzedowski-Calderón and Villa-Salvador [RCV] use the method of Scholz and Reichardt they do not try to bound the number of ramified primes. Indeed for a given l-group G with l q and ζl / ∈ Fq, they construct a Galois extension L of Fq(t) such that G(L/Fq(t)) ∼= G and the genus of L is large. Here and in the sequel, t is a transcendental element over the base field. By the Hurwitz-Riemann genus formula, this means that the number of primes of Fq(t) which ramify in L is also large. The goal of this work is to use the method of Scholz and Reichardt to realize each l-group G over an arbitrary global field with bounded ramification. Theorem A: LetK be a global field and let l be a prime number such that l 6= char(K) and ζl / ∈ K. Then there exists a nonnegative integer r = r(K) such that for each