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

~ 1941

MONODROMIE GROUP

379

of k, whether or not the coefficients of 4(k, zi,..., Zn) involve irrational constants.
Let R be the domain defined by k and the coefficients of
the powers of z in F(z, k). Any rational function 4 of the
roots with coefficients in R, which equals a quantity in R,
equals a rational function of k and hence is unaltered by every
substitution of M. Then, by the Corollary in ~ 149, M is a
subgroup of the Galois group G for R of the equation F=0.
Moreover, M is an invariant subgroup of G. For, let P be
a rational function of the roots with coefficients in R which
belongs to the subgroup M of index v under G. Then 4 is a
root of an equation E of degree v with coefficients in R, so that
) is an algebraic function of k. But 0 possesses monodromie
with respect to k. Hence 4 is a rational function f(k) of k
with perhaps irrational coefficients. Replace the coefficients
of f(k) by independent variables and substitute the resulting expression in place of 4 in E, and let the result be an
identity in k. We obtain certain algebraic equations which the
variable coefficients of f(k) must satisfy. Adjoin to R all of
the roots of these numerical equations. Since 0 is in the enlarged domain, G reduces to a subgroup of M, necessarily M
itself, since the group of monodromie is evidently unaltered
by the adjunction of constants. But the adjunction of all of
the roots of a second equation reduces the Galois group of the
first equation to an invariant subgroup (~ 167). A number of
such adjunctions reduced G to M; whence M is invariant in G.
For example, the Galois group for R(k) of z3-2k:=0 is the symmetric
group G6, since the equation is irreducible in R and the product of the differences of its roots is 6V-3k3. The only circuits causing a permutation
of the roots are those around the origin. Hence M is the cyclic group of
order 3 and is invariant in G.
195. Applications of Monodromie. Jordan * employed
monodromie to determine the Galois group of the equation for
the n-section of the periods of elliptic t and hyperelliptic
functions with 2p periods. For the case of the trisection of
* Traite des Substitutions, pp. 337-369.
t Cf. H. Weber, Elliptische Functionen, 1891, p. 219.