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 𝔡 and the class of 𝔡 is a square in the ideal class group of K. We prove that if 𝔭₁,...,𝔭ₙ are finite primes of K such that ∙ the class of $𝔭_i$ is a square in the ideal class group of K for every i ∈ {1,...,n}, ∙ -1 is a local square at $𝔭_i$ for every nondyadic $𝔭_i ∈ {𝔭₁,...,𝔭ₙ}$, then {𝔭₁,...,𝔭ₙ} is the wild set of some self-equivalence of the field K.
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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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