In 1637 the Toulouse lawyer Pierre de Fermat enunciated a conjecture in number theory which denied the possibility of dividing the nth power of an integer into the sum of two nth integer powers when natural exponents greater than or equal to three are considered. This is the famous "Fermat's last conjecture", now elevated to a theorem thanks to the proof given by Prof. A. Wiles in 1995. In other words, the Diophantine equation x^n + y^n = z^n with n ≥ 3 does not admit integer, primitive and non-trivial solutions. In my article, I set out my observations on a particular case of this theorem. The "admirable proof" which Fermat claimed to have was never found, except for the one in which n = 4 and which the French genius demonstrated with a new method coined by himself and known as the "method of infinite descent". The latter follows the same logic as the already known "ab absurdum reasoning" even if it is articulated with slightly more demanding algebraic procedures.In this article I report my proof of the case n = 4 with the use of an elementary algebraic procedure which, avoiding Fermat's "infinite descent", brings the equation in question back to that of the well-known quadratic equation:x^2 + 2y^2 = z^2 for which all the primitive, non-trivial solutions in the set of integers have already been parameterized. The peculiarity of my method lies in transforming Fermat's quartic equation into a quadratic equation well studied by number theorists and whose solutions are already known. With this strategy it is possible to prove the thesis.