MTL105: Algebra - Introduction class

The following books are recommended for Algebra course -

1. Topics in Algebra by I.N. Herstein
2. Contemporary Abstract Algebra by Joseph A. Gallian

The course mainly deals with -

1. Group Theory
2. Ring Theory
3. Field Theory

Preliminaries

Much of abstract algebra involves properties of integers and sets.

Definitions - Set

A set is a collection of well-defined distinct objects. for e.g. :

$$\mathbb{Z} = \left \{0, \pm 1, \pm 2, ... \right \}$$

An important property of the integers is the well-ordering principle.

Well-Ordering Principle

Every non-empty set of positive integers contains a smallest member.

If we had included negative numbers also, then establishing the notion of smallest number would not have been possible.

The concept of divisibility also plays a fundamental role in the number theory. We write $t \mid s$, i.e. t divides s, if $s = tu$.

Division Algorithm

Let a and b be integers with b > 0. Then there exist unique integers q and r with the property that: $a = bq + r, 0\leq r < b$

Definitions - Greatest Common Divisor

Two numbers a and b are said to have a GCD equal to d if -

• $d \mid a$ and $d \mid b$
• If $e \mid a$ and $e \mid b$ => $e \mid d$

Question : What is gcd(0, 0) = ?

Definitions - Prime Number

A positive integer greater than 1 whose only positive divisors are 1 and the number itself is said to be prime.

Two numbers a and b are said to be relatively prime if $gcd(a, b) = 1.$

GCD is a Linear Combination

For $a \neq 0, b \neq 0$, there exists integers s and t such that $gcd(a, b) = as + bt$. Moreover, gcd(a, b) is the smallest positive integer of the form $as + bt$.

Euclid’s Lemma p|ab implies p|a or p|b

If p is a prime that divides ab, then p divides a or p divides b.

Proof:

Let $p \nmid a$, i.e. p does not divide a. We must show that p divides b. Since p does not divide a, $gcd(p, a) = 1$.

So, we can write $1 = as + pt$. Multiplying both sides by b, $b = abs + ptb$, and since p divides the right-hand side of this equation, thus p divides b also.

• It was mentioned that there are 6 different proofs to show that there are infinitely many primes.

Primes are the building blocks of all integers. The following property convinces us so.

Fundamental Theorem of Arithmetic

Every integer greater than 1 is either a prime or a product of primes. This product is unique, except for the order in which the factors appear. That is, if $n = p_{1}p_{2}...p_{r}$ and $n = q_{1}q_{2}...q_{s}$, where the p’s and q’s are primes, then $r = s$ and, after renumbering the q’s we have $p_{i} = q_{i}$ for all i.

Definitions - Least Common Multiple

Two nonzero integers a and b are said to have s as their LCM if

• $a \mid s$ and $b \mid s$.
• If $a \mid t$ and $b \mid t$ => $s \mid t$.

Mathematical Induction

1. First Principle of Mathematical Induction

Let S be a set of integers containing a. Suppose S has the property that whenever some integer $n \geq a$ belongs to S, then the integer $n + 1$ also belongs to S. Then, S contains every integer greater than or equal to a.

2. Second Principle of Mathematical Induction

Let S be a set of integers containing a. Suppose S has the property that n belongs to S whenvever every integer less than n and greater than or equal to a belongs to S. Then, S contains every integer greater than or equal to a.