On the Size of the Wild Set

Abstract

Abstract. To every pair of algebraic number fields with isomorphic Witt rings one can associate a number, called the minimum number of wild primes. Earlier investigations have established lower bounds for this number. In this paper an analysis is presented that expresses the minimum number of wild primes in terms of the number of wild dyadic primes. This formula not only gives immediate upper bounds, but can be considered to be an exact formula for the minimum number of wild primes.

Wild Primes of a Higher Degree Self-Equivalence of a Number Field

“…In the case of a Hilbert equivalence which is not tame it is natural to ask what the wild set of this equivalence could be. In [13] Somodi estimated the size of the wild set of Hilbert equivalence of degree 2, whereas in [14] and [15] he gave a description of the wild set of Hilbert self-equivalence of degree 2 of the rational number field Q as well as the Gaussian field Q(i). Somodi's results were generalized to a wider family of number fields in [7].…”

A characterization of the finite wild sets of rational self-equivalences

“…In [5] it is proved that any correspondence of defect δ can be extended to a Hilbert symbol equivalence that has no less than δ +|rk 2 C K (S)−rk 2 C L (S ′ )| additional wild primes. When K = L = Q, the second term in the above formula is 0, so any such correspondence can be extended to a rational self-equivalence that has δ additional wild primes.…”

“…Note that ν S (−1) = (−1, −1, 1), ν S (2) = (1, 2, u q ), and ν S (q) = (1, 5, π q ) are mapped by t S to (−1, −1, 1) = ν S ′ (−1), (1,5, π q ) = ν S ′ (q), and (1, 2, u q ) = ν S ′ (2) respectively, so the defect of this correspondence is equal to 0. Therefore, the correspondence can be extended tamely to a rational self-equivalence denoted by (t (1) , T (1) ).…”

“…Any correspondence between two Hilbert symbol equivalent number fields can be extended to a Hilbert symbol equivalence that has a finite wild set ( [5]). …”

“…To any correspondence one associates a non-negative integer, called the defect of the correspondence ( [5]). Specifically, if t S is the direct product of the maps t P with P ∈ S and…”