casus irreducibilis


Let the polynomialPlanetmathPlanetmath

P(x):=xn+a1xn-1++an

with complex coefficients aj be irreducible (http://planetmath.org/IrreduciblePolynomial2), i.e. irreducible in the field (a1,,an)  of its coefficients.  If the equation  P(x)=0  can be solved algebraically (http://planetmath.org/AlgebraicallySolvable) and if all of its roots are real, then no root may be expressed with the numbers aj using mere real radicalsPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/NthRoot) unless the degree (http://planetmath.org/AlgebraicEquation) n of the equation is an integer power (http://planetmath.org/GeneralAssociativity) of 2.

References

  • 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet.  Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).
Title casus irreducibilisMathworldPlanetmath
Canonical name CasusIrreducibilis
Date of creation 2013-03-22 15:21:00
Last modified on 2013-03-22 15:21:00
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Theorem
Classification msc 12F10
Related topic RadicalExtension
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Related topic TakingSquareRootAlgebraically
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