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• W1570243798 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>f(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a monic polynomial in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Z left-bracket x right-bracket> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {Z}[x]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with no rational roots but with roots in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Q Subscript p> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Q</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>mathbb {Q}_{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, or equivalently, with roots mod <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is known that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>f(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> cannot be irreducible but can be a product of two or more irreducible polynomials, and that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>f(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a product of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m greater-than 1> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>m>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> irreducible polynomials, then its Galois group must be a union of conjugates of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=application/x-tex>m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> proper subgroups. We prove that for any <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m greater-than 1> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>m>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, every finite solvable group that is a union of conjugates of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=application/x-tex>m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m equals 2> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>m=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Q left-parenthesis t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Q</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {Q}(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W1570243798 date "2008-02-12" @default.
• W1570243798 modified "2023-09-23" @default.
• W1570243798 title "Polynomials with roots in ℚ_{𝕡} for all 𝕡" @default.
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• W1570243798 doi "https://doi.org/10.1090/s0002-9939-08-09155-7" @default.
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