318 7. FIELD EXTENSIONS – FIRST LOOK radicals and, in doing so, radically changed the nature of algebra. Galois associated with a polynomial p.x/ over a ﬁeld K a canonical smallest ﬁeld L containing K in which the polynomial has a complete set of roots and, moreover, a canonical group of symmetries of L , which acts on the roots of p.x/ . Galois’s brilliant idea was to study the polynomial equation p.x/ D0 by means of this symmetry group. In particular, he showed that solvability of the equation by radicals corresponded to a certain property of the group, which also became known as solvability . The Galois group associated to a polynomial equation of degree n is always a subgroup of the permutation group S n . It turns out that subgroups of S n for n ± 4 are solvable, but S 5 is not solvable. In Galois’s theory, the nonsolvability of S 5 implies the impossibility of an analogue of the quadratic formula for equations of degree 5.
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