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The generalized Jacobi polynomials are a class of polynomials that show up frequently in mathematics and have broad applications, even in areas outside of mathematics. This project considers the arithmetic and algebraic properties of a particular specialization of the Jacobi polynomials; that is, we fix one variable and look at the resulting curve defined by each polynomial. We show that each polynomial is irreducible, and that its Galois group is the symmetric group on n letters, where n is the degree of the polynomial. To be more precise, the polynomials we are considering are in two variables, but we can treat one variable as a coefficient which we intend to specialize, i.e. substitute in a rational value for. We further show that there are only finitely many rational specializations which produce a reducible polynomial over the rational numbers, and we also characterize the sets of specializations which have smaller Galois group, again over the rational numbers.
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Etropolski, Anastassia, "Specializations of Jacobi polynomials and their algebraic properties" (2011). Senior Projects Spring 2011. 196.
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