Title: A constructive theory for extensions of p-adic fields

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English title: Symmetric functions and Newton polynomials

Italian title: Funzioni simmetriche e polinomi di Newton

In my thesis we present the classical material of the theory of symmetric functions, with applications to the study of the characters of the symmetric group. Furthermore, we study the problem of deciding when a collection of Newton polynomials (or power sums) generate the field of rational symmetric functions. In particular, we present the following origin result: in positive characteristic p the polynomials N_a=x^a+y^a, N_b=x^b+y^b, N_c=x^c+y^c generate the symmetric field in x,y whenever a,b,c are relatively prime integers such that a,b,c,a-b,a-c,b-c are prime to p. We also give a counterexample showing that it is necessary to require that the differences are prime to p.

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• the main document (in italian),
• the presentation (in italian),
• and the source code (of the thesis and the presentation).