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• W2091875737 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal O</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an order of an algebraic number field. It was shown by Ge that given a factorization of an <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal O</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-ideal <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German a> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>a</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {a}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into a product of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal O</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-ideals it is possible to compute in polynomial time an overorder <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O prime> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>mathcal O’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal O</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a <italic>gcd-free</italic> refinement of the input factorization; i.e., a factorization of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German a script upper O prime> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>a</mml:mi> </mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {a}mathcal O’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into a power product of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O prime> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>mathcal O’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-ideals such that the bases of that power product are all invertible and pairwise coprime and the extensions of the factors of the input factorization are products of the bases of the output factorization. In this paper we prove that the order <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O prime> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>mathcal O’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the smallest overorder of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal O</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in which such a gcd-free refinement of the input factorization exists. We also introduce a partial ordering on the gcd-free factorizations and prove that the factorization which is computed by Ge’s algorithm is the smallest gcd-free refinement of the input factorization with respect to this partial ordering." @default.
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• W2091875737 title "On factor refinement in number fields" @default.
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• W2091875737 doi "https://doi.org/10.1090/s0025-5718-99-01023-6" @default.
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