From the Pell's equation to Gross-Stark units

This thesis is a mathematical journey from the Pell’s equation to Gross–Stark units, centered around the theme of the relationship between leading terms of L-series, and algebraic units. Our story begins with Pell’s equation, and two methods to solve it. In particular, we will focus on the study the fundamental solution of a real quadratic number field k. Then we will move to a general abelian extension of number fields K/k by stating the Stark conjecture. To conclude we will discuss the p-adic analogue of the Stark conjecture, namely the Gross–Stark conjecture. We will state the conjecture for k a real quadratic number field and K its narrow Hilbert class field. We will define the Gross–Stark unit, and compute an explicit example