Abel-Ruffini's theorem
The set of Abel-Ruffini stating that a general polynomial equation fifth or higher degree not d by radicals. H. Root expressions , is solvable.
history
Paolo Ruffini , Teoria generale delle equazioni , 1799
The first proof of this theorem was published by Paolo Ruffini in 1799. However, this evidence was sketchy and largely ignored. A complete proof was provided by Niels Henrik Abel in 1824 .
The Galois theory, developed a little later by Évariste Galois , provides a deeper insight into the problem . Using the more general results of Galois theory, only two points need to be shown to prove Abel-Ruffini's theorem:
- The general fifth degree equation (i.e. the equation with variables as coefficients) has the symmetrical group S 5 as the Galois group
- The symmetrical group S 5 cannot be resolved because it contains the alternating group A 5 of order 60 as the only real normal divisor , and this is simple and not of prime order .
literature
- Jörg Bewersdorff : Algebra for Beginners: From Equation Resolution to Galois Theory , Springer Spectrum, 5th Edition 2013, ISBN 978-3-658-02261-7 , doi : 10.1007 / 978-3-658-02262-4 .
- Peter Pesic: Abel's proof . Jumper. Berlin et al. 2005, ISBN 3-540-22285-5 , doi : 10.1007 / 978-3-540-27309-7 .
- Jean-Pierre Tignol: Galois' Theory of Algebraic Equations . Reprint. World Scientific, Singapore et al. 2004, ISBN 981-02-4541-6 , doi : 10.1142 / 9789812384904 .