Number Theory CST 292 KTU Honors IV Semester - Dr Binu V P

 

Divisibility  We say that a nonzero $b$ divides $a$ if $a=qb$ for some $m$, where $a$,$b$ and $q$ are integers.That is $b$, divides $a$ if there is no remainder on division.The notation $b \mid a$ is commonly used to mean$b$ divides $a$.Also, if  $b|a$, we say that $b$ is a divisor of $a$. If $b$ does not divide $a$ we denote it by $b \nmid a$. For example, $6 \mid 36$  but $7 \nmid 36$. Note that $b \mid 0$ for any non-zero integer $b$ and $1\mid a$ for any integer $a$. When we write $b \mid a$, it is tacitly assumed that $b$ is a non-zero integer. We can easily deduce the following properties form the definition of divisibility itself Let $a, b, c ,d$ be any non-zero integers. 1.if $a|1$ , then $a = \pm 1$ 2. If $a | b$ and $b|c $  then  $a |c$. Proof: (1) Suppose $b = na$ and $c = mb$, where $n; m \in Z$. Then $c = m(na) = (mn)a$ and so $a | c$. 3. If $a | b$ and $c| d$ then $ac | bd.$ Proof: (2) Suppose $b = na$ and $d = mc$ where $n; m \in Z.$ Then $bd = (na)(mc) = (mn)(ac)$, i

  Groups : A  group  is a mathematical structure that consists of a set of elements along with an operation (usually called  addition  or  multiplication ). Key properties of a group: Closure: The result of the operation on any two elements in the group remains within the group. Associativity: The operation is associative (i.e., the order of performing the operation doesn’t matter). Identity element: There exists an element (often denoted as  e ) such that combining it with any other element doesn’t change the other element. Inverse element: Each element has an inverse (opposite) element within the group. Example: The integers under addition form a group. Rings : A  ring  is a more general structure than a group. It includes both  addition  and  multiplication  operations. Key properties of a ring: Closure under addition and multiplication. Associativity of both operations. Distributive property: Multiplication distributes over addition. Unlike groups, rings need not have multiplicativ

Groups, rings, and fields are the fundamental elements of a branch of mathematics known as abstract algebra, or modern algebra. In abstract algebra, we are concerned with sets on whose elements we can operate algebraically; that is, we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set. These operations are subject to specific rules, which define the nature of the set. By convention, the notation for the two principal classes of operations on set elements is usually the same as the notation for addition and multiplication on ordinary numbers. Let's break down the concepts of groups, rings, and fields in the context of number theory, and then we'll discuss the notion of group rings and fields. 1. Groups: In mathematics, a group is a set equipped with an operation that combines two elements to produce a third, satisfying four fundamental properties: Closure: The operation combines elements to produce another element in the set.

In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered by its "natural" or "magnitude" order in which $x$ precedes $y$ if and only if  $y$ is either $x$ or the sum of $x$ and some positive integer (other orderings include the ordering $2,4,6$; and $1,3,5$). The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem". On other occasions it is understood to be the proposition that the set of integers $\{…,−2,−1,0,1,2,3,…\}$ contains a well-ordered subset, called the natural numbers, in which every nonempty subset contains a least element. Well-ordering Principle: If $S$ is a non-empty set of non-negative integers, then $S$ has a least element, i.e., there is an integer $c \in S$ such that $c \le x$ for all $ x \in S$. The principle of mathematical induction follows direct

Modular arithmetic is a fascinating branch of mathematics that deals with numbers “wrapping around” when they reach a certain value, known as the modulus. Here are the key concepts: Definition: In modular arithmetic, we work with integers and consider their remainders when divided by a fixed quantity (the modulus). Think of it like a clock: after reaching 12 hours, the clock “wraps around” to 1. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss The Modulus If $a$  is an integer and $n$ is a positive integer, we define $ a \: mod \: n $ to be the remainder when $a$ is divided by $n$ . The integer is called the modulus. Thus, for any integer , $$a = qn+r \quad  0 \le r \lt n \quad q=\lfloor a/n \rfloor$$ $$r= a - \lfloor a/n \rfloor .n $$ Example: $11 \: \mod \: 7 =4$ and $-11 \: \mod  \: 7=3$ Two integers $a$ and $b$ are  said to congruent modulo  $n$, if $a \: \mod \: n= b \: \mod \: n$.This is written as $$a \equiv b (\mod  n)$$ Example: $5 \equiv 16 (

Bézout’s identity, also known as Bézout’s lemma, is a fundamental theorem in elementary number theory. Named after Étienne Bézout, who proved it for polynomials, the identity states the following: Given two nonzero integers $a$ and $b$, let $d$ be their greatest common divisor (GCD). Then there exist integers $x$ and $y$ such that: $$ax + by = d$$ In other words, Bézout’s identity guarantees that we can express the GCD of $a$ and $b$ as a linear combination of these integers. Furthermore, the integers $x$ and $y$ are not unique; there are infinitely many pairs that satisfy this equation. $$ax+by=d$$ $$ax+ab+by-ab=d$$ $$a(x+b)+b(y-a)=d$$ In general $x+nb$ and $y-na$ can be done for any value of $n$. So $x$ and $y$ are not unique. Key points about Bézout’s identity: The integers $x$ and $y$ are called Bézout coefficients for the pair $(a, b)$. These coefficients can be computed using the extended Euclidean algorithm. Bézout’s identity has applications in various areas of number theory