Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

320

EQUATIONS SOLVABLE BY RADICALS [CH. XVI

In fact, by adjoining one root xl of F2(x)=O, we adjoin
all of its roots, since each is a rational function of xl with coefficients in R (~ 155). For, the identity is the only substitution
of the group for R of F2(x)=0 which leaves xl numerically
unaltered.
We shall state certain results not presupposed in what follows. A
brief argument (cf. Dickson's Theory of Algebraic Equations, 1903, p. 83)
now leads to Abel's theorem: The roots of an equation solvable by radicals
can be given such a form that each of the radicals occurring in the expressions for the roots are expressible rationally in terms of the roots of the
equation and certain roots of unity. This was proved by Abel by a long
algebraic discussion without the aid of groups and employed in his proof
of the impossibility * of solving by radicals the general equation of degree
n>5.
For a domain R an irreducible equation of. prime degree whose roots
are all rational functions of two of the roots with coefficients in R is called
a Galoisian equation. Galois proved that it is solvable by radicals and that
every irreducible equation of prime degree which is solvable by radicals
is a Galoisian equation. For a detailed exposition with illustrative examples, see Dickson's Theory of Algebraic Equations, 1903, pp. 87-93.
A cubic equation having three real roots cannot t be solved by real
radicals (the " irreducible case ").
* Cited in ~ 166. Cf. Serret, Cours d'Algebre superieure, ed. 4, vol. 2, pp.
497-517.
t H. Weber, Algebra, ed. 2, 1898, vol. 1, 657; Kleines Lehrbuch der Algebra,
1912, p. 381.