History: Gauss fields Fields with p elements with p prime. Finite fields: existence, unicity, structure, explicit construction. Cyclotomic polynomials. Frobenius automorphisms. Galois's Theory of finite fields. Error correcting codes. Construction of Irreducible polynomials over finite fields Factorization of polynomials over finite fields Permutation Polynomials/Chebicev Polynomials Exponential sums over finite fields The discrete logarithm", - the Carleman inequalities approach (see [2]), - and I will conclude with a recent technic using pointwise observations. (see [3]). References [1] M Chouli. Une introduction aux problemes inverses elliptiques et paraboliques. Springer-Verlag, 2009. [2] M Yamamoto. Carleman estimates for parabolic equations and applications. Inverse Problems, 25(12):123013, 2009. [3] L Roques and M Cristofol. On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation. Nonlinearity, 23:675 -- 686, 2010.