*by Ganapathy Sundaram*

Consider a polynomial in two variables *X* and *Y*. This
can be treated as a polynomial of degree *n* in *Y*, whose
coefficients are polynomials in *X*. Now plug in values for
*X*, and it can be seen that for *most values* of
*X*, we get *n*-corresponding roots for *Y*. But
there are some places where there are fewer than *n*-roots for
*Y*. These special places are called *discriminant
points*. **If there are no discriminant points, then f factors
completely.** This was noticed by the famous mathematician
Riemann in the mid-1800's by shrinking the plane to a point. Prior to
Riemann, the famous French mathematician Galois made the same observation
by looking at permutations of the roots of the equations. The set of all
permutations of the roots forms a group and is now referred to as the

Abhyankar's work in Galois theory started with his Harvard thesis in the
early 1950's under the able guidance of the famous mathematician, Oscar
Zariski. He was interested in polynomials in three variables (i.e.,
surfaces), and he was working on removing so-called singularities
(beak-like points) of the surface over modular fields (i.e., fields
obtained by fixing a prime number and considering remainders of every
integer after dividing by that fixed prime). This problem also has its
origins in Riemann's work, but the three variable case over "usual"
fields (i.e., complex numbers) was solved by Jung in 1908. In his
thesis, Abhyankar first showed that over modular ground fields, Jung's
classical method of changing the surface to get rid of singularities does
not work because of *strange properties* of Galois groups. In
particular, he discovered that **things were different over modular
ground fields because a polynomial with coefficients in a modular field
need not factor completely even if it has no discriminant
points.** As a result, in 1957 he launched a systematic attack on
understanding Galois groups of polynomials (in two variables and later in
more variables) and conjectured what Galois groups can be obtained over
modular ground fields. Really what he wrote down was more than a
conjecture-he laid out a philosophy. What Harbater and Raynaud have
proved are special cases of this philosophy pertaining to polynomials in
two variables, i.e., curves. This is only the tip of the iceberg, but
the excitement is due to the fact that one now has a structured approach to
understanding Galois groups of various equations.