Answer the following question: Information: - 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. - "Not to be confused with his wife Michele Raynaud, who is also a French mathematician working in algebraic geometry." - The term proposition has a broad use in contemporary philosophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other "propositional attitudes" (i.e., what is believed, doubted, etc.), the referents of that-clauses, and the meanings of declarative sentences. Propositions are the sharable objects of attitudes and the primary bearers of truth and falsity. This stipulation rules out certain candidates for propositions, including thought- and utterance-tokens which are not sharable, and concrete events or facts, which cannot be false. - In abstract algebra , Abhyankar 's conjecture is a 1957 conjecture of Shreeram Abhyankar , on the Galois groups of algebraic function fields 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 and David Harbater . The problem involves a finite group G , a prime number p , and the function field of nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p. The question addresses the existence of Galois extensions L of K ( C ) , with G as Galois group , and with restricted ramification . From a geometric point of view L corresponds to another curve C , and a morphism : C C. Ramification geometrically , and by analogy with the case of Riemann surfaces , consists of a finite set S of points x on C , such that restricted to the complement of S in C is an etale morphism . 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 . The subgroup p ( G ) is defined to be the subgroup generated by all the Sylow subgroups 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 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 of C. Then G can be realised if and only if n s + 2 g. - Algebra (from Arabic ""al-jabr"" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. - In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a list of axioms. - In mathematics, a polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate is . An example in three variables is . - In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been known to have infinitely many solutions since antiquity. - In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field that contains the base field and satisfies additional properties. For instance, the set Q(2) = {"a" + "b"2 | "a", "b" Q} is the smallest extension of Q that includes every real solution to the equation "x" = 2. - In mathematics, more specifically in the area of modern 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 polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Evariste Galois who first discovered them. - A mathematician is someone who uses an extensive knowledge of mathematics in his/her work, typically to solve mathematical problems. - In mathematics, more specifically in abstract algebra, Galois theory, named after Evariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood. - The University of Pennsylvania (commonly known as Penn or UPenn) is a private, Ivy League university located in Philadelphia, Pennsylvania, United States. Incorporated as "The Trustees of the University of Pennsylvania", Penn is one of 14 founding members of the Association of American Universities and one of the nine original colonial colleges. - Evariste Galois (25 October 1811 31 May 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem standing for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. He died at age 20 from wounds suffered in a duel. - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous observations that cannot satisfactorily be explained with the available scientific theories. Even though the words "hypothesis" and "theory" are often used synonymously, a scientific hypothesis is not the same as a scientific theory. A working hypothesis is a provisionally accepted hypothesis proposed for further research. - In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term "abstract algebra" was coined in the early 20th century to distinguish this area of study from the other parts of algebra. - In mathematics, a conjecture is a conclusion or proposition based on incomplete information, for which no proof has been found. Conjectures such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (which was a conjecture until proven in 1995) have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. - Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. - In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . It was proposed by , after whom it is named. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. Given the paragraphs above, decide what entity has the relation 'instance of' with 'theorem'.
Answer:
abhyankar's conjecture