There are many open questions in number theory branch of mathematics. Some of them are related with real quadratic number fields one of the types of quadratic fields. (Let Q(α) be a number field where α is of degree 2. These fields are known as Quadratic number fields. If α= √d and d is positive square free integer, then Q(α) is called as real quadratic number field)
The class number of a real quadratic field is one of the significant and basic tool to determinate structure of the field in number theory. A lot of mathematicians have worked on this topic seriously since Gauss. There are many types of methods to calculate class number for real quadratic fields. One of them is the Dirichlet’s class number h(d) formula defined by regulator, dicriminant and L- function. It is clear that these tools such as fundamental unit, discriminant etc are crucial for the real quadratic fields. Also, we know that any periodic / pur periodic continued fraction defines a real number which is quadratic over Q. So, it is seen that there is a connection between continued fraction expansions and the theory of real quadratic number felds.
The aim of this paper is to get explicit and theoretical results on particular real quadratic number fields. In this paper, we are interested in the fundamental unit and continued fraction expansions for such real quadratic number fields for d>0,d≡1 (mod4). Significant and useful results are obtained in this work for real quadratic number theory.
Anahtar Kelimeler: Real Quadratic Fields, Continued Fraction Expansions, Fundamental Units, Algebraic Number Theory Computations.