Construction of Universal p-adic Fields

This thesis focuses on the construction of universal p-adic fields, defined as complete and algebraically closed extensions of Qp. We introduce the ring of p-adic integers Zp and its field of fractions Qp, also obtained as the completion of Q with respect to the p-adic valuation. Since the algebraic closure Qp^a of Qp is not complete, we consider its completion Cp. As Cp is not spherically complete, we use ultrafilters to construct Ωp, a spherically complete and algebraically closed extension of Qp^a. Within this framework, Cp is defined as the topological closure of Qp^a in Ωp.