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The image of the natural homomorphism of Witt rings of orders in a global field

100%
Rothkegel B.
Acta Arithmetica
|
2013
|
tom 160
|
nr 4
349-384
EN
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.
2
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Wild primes of a self-equivalence of a number field

63%
Czogała A., Rothkegel B.
Acta Arithmetica
|
2014
|
tom 166
|
nr 4
335-348
EN
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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Wild Primes of a Higher Degree Self-Equivalence of a Number Field

51%
Czogała A., Rothkegel B., Sładek A.
Annales Mathematicae Silesianae
|
2016
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tom 30
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nr 1
17-38
EN
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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