We aim in this research to define the separable and inseparable extensions. This is achieved by introducing the basic concepts of field theory where the most important one is the field that plays a major role in algebra since it provides a useful generalization of many number systems such as real numbers, complex numbers, rational numbers and many other systems. A field is an algebraic structure that is indispensable to much of mathematics. For instance, in algebraic geometry, number theory and calculus. In order to delve deeper into our goal, we have proved that any irreducible polynomial has a root in some extension which support having n roots in an extension of any polynomial of degree n, ending up with the separable extension that has distinct roots of any polynomial of degree n. Furthermore, we have solved the three geometric problems by proving that some of the real numbers are not constructible.