The Monodromy Group of an Algebraic Function

This Demonstration shows the structure of the branches of a multivalued function

w(z)

defined by a polynomial equation

P(w,z)=0

, illustrating the transitions between the branches along paths going around a branch point. The actual configuration may depend on the choice of the branch cuts, but the group generated by the branch cycles is always the same. In general this group is a normal subgroup of the Galois group of

P(w,z)

over



. A number of important properties of

w(z)

can be inferred from the structure of the monodromy group:

•

P(w,z)

is absolutely irreducible if and only if the group is transitive;

• if

P(w,z)

is irreducible,

w(z)

can be expessed in radicals as a function of

z

if and only if the group is solvable;

• the genus of

P(w,z)

can be computed from the branch cycles using the Riemann–Hurwitz formula. If

P(w,z)

is irreducible and the genus is zero, the integral of

w(z)

can always be expressed

in terms of

w(z)

and elementary functions. (The converse is not true: it is possible for

w(z)

to have an elementary antiderivative if the genus is greater than zero.)