Splitting fields of real irreducible representations of finite groups

<p>We show that any irreducible representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi>ρ<! -- ρ --></mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a finite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of exponent <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>, realisable over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is realisable over the field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper E colon-equal double-struck upper Q left-parenthesis zeta Subscript n Baseline right-parenthesis intersection double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>≔</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>ζ<! -- ζ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∩<! -- ∩ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">E≔\mathbb {Q}(\zeta _n)\cap \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of real cyclotomic numbers of order <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>, and describe an algorithmic procedure transforming a realisation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi>ρ<! -- ρ --></mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="double-struck upper Q left-parenthesis zeta Subscript n Baseline 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:msub> <mml:mi>ζ<! -- ζ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}(\zeta _n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to one over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. </p>