We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed by radicals, i.e., by the operations $+,-,·,:$, and $\sqrt[n]{·}$. Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals.

EUDML-ID : urn:eudml:doc:287786
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Mots clés:

@article{702970,
     title = {Why quintic polynomial equations are not solvable in radicals},
     booktitle = {Application of Mathematics 2015},
     series = {GDML\_Books},
     publisher = {Institute of Mathematics CAS},
     address = {Prague},
     year = {2015},
     pages = {125-131},
     zbl = {06669924},
     url = {http://dml.mathdoc.fr/item/702970}
}

Křížek, Michal; Somer, Lawrence. Why quintic polynomial equations are not solvable in radicals, dans Application of Mathematics 2015, GDML_Books,  (2015), pp. 125-131. http://gdmltest.u-ga.fr/item/702970/