abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...

, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...

s of

algebraic function field In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebrai ...

s of characteristic ''p''. The soluble case was solved by Serre in 1990 and the full conjecture was proved in 1994 by work of

Michel Raynaud Michel Raynaud (; 16 June 1938 – 10 March 2018 Décès de Michel Raynaud
So ...

and

David Harbater David Harbater (born December 19, 1952) is an American mathematician at the University of Pennsylvania, well known for his work in Galois theory, algebraic geometry and arithmetic geometry. Early life and education Harbater was born in New York ...

.. The problem involves a finite group ''G'', a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

''p'', and the function field ''K(C)'' of a nonsingular integral

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

''C'' defined over an

algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''K'' of characteristic ''p''. The question addresses the existence of a

Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...

''L'' of ''K''(''C''), with ''G'' as Galois group, and with specified ramification. From a geometric point of view, ''L'' corresponds to another curve ''C''′, together with a morphism :π : ''C''′ → ''C''. Geometrically, the assertion that π is ramified at a finite set ''S'' of points on ''C'' means that π restricted to the complement of ''S'' in ''C'' is an

étale morphism In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy t ...

. This is in analogy with the case of

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...

s. In Abhyankar's conjecture, ''S'' is fixed, and the question is what ''G'' can be. This is therefore a special type of

inverse Galois problem In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There ...

. The subgroup ''p''(''G'') is defined to be the subgroup generated by all the

Sylow subgroup In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...

s of ''G'' for the prime number ''p''. This is a normal subgroup, and the parameter ''n'' is defined as the minimum number of generators of :''G''/''p''(''G''). Then for the case of ''C'' the

projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...

over ''K'', the conjecture states that ''G'' can be realised as a Galois group of ''L'', unramified outside ''S'' containing ''s'' + 1 points, if and only if :''n'' ≤ ''s''. This was proved by Raynaud. For the general case, proved by Harbater, let ''g'' be the

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...

of ''C''. Then ''G'' can be realised if and only if :''n'' ≤ ''s'' + 2 ''g''.

References

External links

* {{MathWorld, urlname=AbhyankarsConjecture, title=Abhyankar's conjecture
A layman's perspective of Abhyankar's conjecture
from

Purdue University Purdue University is a public land-grant research university in West Lafayette, Indiana, and the flagship campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and mone ...

Algebraic curves Galois theory Theorems in abstract algebra Conjectures that have been proved