Reciprocity law of quadratic extensions

Abstract

In the first chapter, basic definitions and results which will be used in the following chapters of this thesis are presented. In the following chapter, ideal classes and classes of quadratic forms are reviewed. Then the relationship between the ideal classes of the quadratic field Q(pD) with discriminant and the classes of quadratic forms having discriminant is established. It is proved that if two forms are equivalent, then they are constructed by two equivalent ideals and conversely equivalent ideals construct equivalent forms. The next chapter aims to present one of the proofs of the quadratic reciprocity law which is based on the theory of quadratic number fields. Instead of developing the theory of binary quadratic forms, a proof using the ideal theoretic approach is given since the relation between ideals and forms is discussed in the previous chapter. The Hilbert’s symbol for quadratic number fields is defined in this chapter and it is compared with Legendre symbol. Then genus is defined by using character sets and the quadratic reciprocity law is proved. Furthermore, the number of genera is found. The following chapter again aims to prove the quadratic reciprocity law by using the theory of quadratic number fields. But for this chapter, we will first discuss how the strict sense equivalence change the class number. Then, we will find the number of genera by using exact sequences. It is easier than the previous section since considering strict equivalence brings all cases into one case. With these results, again a proof of the quadratic reciprocity law is given. In addition, genus character and genus field with their properties is presented. In the last chapter, quadratic reciprocity law over Q(i) is presented. The proof is based on the theory of Dirichlet number fields. The relative Hilbert symbol is defined for quadratic number fields over Q(i) and the number of genera of a Dirichlet number field is found by using the parallel arguments in Chapter 4. The number of genera again leads us to prove the quadratic reciprocity law over Q(i).