casus irreducibilis

Let the polynomial

P(x):=x^{n}+a_{1}x^{n-1}+\ldots+a_{n}

with complex coefficients $a_{j}$ be irreducible (http://planetmath.org/IrreduciblePolynomial2), i.e. irreducible in the field $\mathbb{Q}(a_{1},\,\ldots,\,a_{n})$ 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