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Let R be a Dedekind domain whose field of fractions is a global field. Moreover, let 𝓞 < R be an order. We examine the image of the natural homomorphism φ : W𝓞 → WR of the corresponding Witt rings. We formulate necessary and sufficient conditions for the surjectivity of φ in the case of all nonreal quadratic number fields, all real quadratic number fields K such that -1 is a norm in the extension K/ℚ, and all quadratic function fields.
Let K be a number field. Assume that the 2-rank of the ideal class group of K is equal to the 2-rank of the narrow ideal class group of K. Moreover, assume K has a unique dyadic prime and the class of is a square in the ideal class group of K. We prove that if ₁,...,ₙ are finite primes of K such that
∙ the class of is a square in the ideal class group of K for every i ∈ 1,...,n,
∙ -1 is a local square at for every nondyadic ,
then ₁,...,ₙ is the wild set of some self-equivalence of the field...
Let ℓ > 2 be a prime number. Let K be a number field containing a unique ℓ-adic prime and assume that its class is an ℓth power in the class group CK. The main theorem of the paper gives a sufficient condition for a finite set of primes of K to be the wild set of some Hilbert self-equivalence of K of degree ℓ.
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