# casus irreducibilis

Let the polynomial^{}

$$P(x):={x}^{n}+{a}_{1}{x}^{n-1}+\mathrm{\dots}+{a}_{n}$$ |

with complex coefficients ${a}_{j}$ be irreducible (http://planetmath.org/IrreduciblePolynomial2), i.e. irreducible in the field $\mathbb{Q}({a}_{1},\mathrm{\dots},{a}_{n})$ 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 ${a}_{j}$ using mere real radicals^{} (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 irreducibilis^{} |
---|---|

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 |

Related topic | CardanosFormulae |

Related topic | TakingSquareRootAlgebraically |

Related topic | EulersDerivationOfTheQuarticFormula |