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# 5: Primitive Roots and Quadratic Residues

• • Contributed by Wissam Raji
• Associate Professor and the Chairman (Mathematics) at American University of Beirut
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In this chapter, we discuss the multiplicative structure of the integers modulo $$n$$. We introduce the concept of the order of integer modulo $$n$$ and then we study its properties. We then define primitive roots modulo $$n$$ and show how to determine whether an integer is primitive modulo $$n$$ or not. We later find all positive integers having primitive roots and prove related results. We define the concept of a quadratic residue and establish its basic properties. We then introduce Legendre symbol and also develop its basic properties. We also introduce the law of quadratic reciprocity. Afterwards, we generalize the notion of Legendre symbol to the Jacobi symbol and discuss the law of reciprocity related to Jacobi symbol.

• 5.1: The order of Integers and Primitive Roots
• 5.2: Primitive Roots for Primes
In this section, we show that every integer has a primitive root. To do this we need to introduce polynomial congruence.
• 5.3: The Existence of Primitive Roots
In this section, we demonstrate which integers have primitive roots. We start by showing that every power of an odd prime has a primitive root and to do this we start by showing that every square of an odd prime has a primitive root.
• 5.4: Introduction to Quadratic Residues and Nonresidues
• 5.5: Legendre Symbol
In this section, we define Legendre symbol which is a notation associated to quadratic residues and prove related theorems.
• 5.6: The Law of Quadratic Reciprocity
Given that p and q are odd primes. Suppose we know whether q is a quadratic residue of p or not. The question that this section will answer is whether p will be a quadratic residue of q or not. Before we state the law of quadratic reciprocity, we will present a Lemma of Eisenstein which will be used in the proof of the law of reciprocity. The following lemma will relate Legendre symbol to the counting lattice points in the triangle.
• 5.7: Jacobi Symbol
In this section, we define the Jacobi symbol which is a generalization of the Legendre symbol. The Legendre symbol was defined in terms of primes, while Jacobi symbol will be generalized for any odd integers and it will be given in terms of Legendre symbol.

## Contributors

• Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution (CC BY) license.