Sections 1.1-1.3

Abstract Algebra Hungerford

Section 1.1: The Division Algorithm


Summary:

Division is the repeated subtraction of a number there is a remaining value that is less than the number. This is idea is formally presented with the Division Algorithm

Let a, b be integers with b > 0. Then there exist unique integers q and r such that

a = bq+r\mbox{ and } 0\leq r< b

My Analysis:

This section is very straight forward and surprisingly clear.  I especially like how Hungerford explains that division is simply “repeated subtraction” (5), this is a remarkably uncomplicated description.

Admittedly I have not done any mathematical proofs in some time, yet Hungerford describes some of the basic methods used for proofs. That is he discusses how to demonstrate that a set is non-empty, proof by contradiction, uniqueness etc.

Section 1.2: Division


Summary:

Hungerford’s formal definition of Division:

Let a and b be integers with b≠0. We say that b divides a (or that b is a divisor of a, or that b is a factor of a) if a=bc for some integer c. In symbols, “b divides a” is written b\mid a and “b does not divide a” is written b \nmid a (page 9).

Some basic facts:

a and -a have the same divisors

Every divisor of the non-zero integer a is less than or equal to |a|

a non-zero integer has a finite number of divisors

The greatest common divisor (gcd) of two integers (of two integers a and b) is defined to be the largest integer d that divides both a and b.

Two integers are said to be relatively prime if the gcd of the two integers is one. For example, (10,9)=1.

Hungerford’s Theorem 1.2

Let a and b be integers, not both 0, and let d be their greatest common divisor. Then there exists (not necessarily unique) integers u and v such that d = au + bv (page 12).

Hungerford’s Corollary 1.3

Let a and b be integers, not both 0, and let d be a positive integer. Then d is the greatest common divisor of a and b if and only if d satisfies these conditions:

(i) d\mid a and d\mid b

(ii) if c\mid a and c\mid b, then c\mid d (page 13)

Hungerford’s Theorem 1.4

If a\mid bc and (a,b)=1, then a\mid c.

My Analysis:

The notation used can be a little daunting, and sometimes it should be written in a more recognizable form that is b|a simply means $latex a=bc$. Similarly the notation (a,b) refers to the greatest common divisor of a and b.

Theorem 1.2 is a bit difficult to understand. Initially it seems that d=(a,b) but Hungerford corrects this by saying that this is not implied. The one example provided (page 12) is 2579\cdot 826+4321\cdot (-493)=1. The integer d should be any integer not limited to 1. Coming up with a unique example is a bit challenging. Say 24 and 18. The greatest common divisor is 6. Writing this in the way defined in Theorem 1.2 we say 6 = 24\cdot m + 18 \cdot n. For this to be true let m=7 and n=-9. Exactly how d=(a,b) isn’t necessarily true as of yet remains to be understood.

Other than it’s use for basic algebra I have yet to correlate these facts to any real use.

Section 1.3: Primes and Unique Factorization


Summary:

Definition of A Prime:

An integer p is prime if p ≠ 0, ±1 and the only divisors of p are ±1 and ±p (page 16).

Hungerford’s Theorem 1.5

Let p be an integer with p ≠ 0, ±1. Then p is prime if and only if p has this property: whenever p\mid bc, then p\mid b or p\mid c (page 17).

Hungerford’s Corollary 1.6

If p is prime and p\mid a_1\cdot a_2\cdot a_3\cdot ...\cdot a_n, then p divides at least one of the a_i (page 17).

Hungerford’s Theorem 1.7

Every integer n except 0, ±1 is a product of primes (page 17).

Hungerford’s Theorem 1.8: The Fundamental Theorem of Arithmetic

All integers (except 0, ±1) are a product of primes.

Hungerford’s Theorem 1.10

Let n > 1, if n has no positive prime factor less than or equal to √n, then n is prime (page 21).

My Analysis:

Theorem 1.10 I find to be super interesting. The number 189 is not prime since 189/3=63. Now √189 = 13.748 yet as was just shown 3 is a factor of 189. Yet 139 is prime, using the theorem √139=11.790 so the only factors we need to worry about are 2, 3, 5, 7. None of which are factors and so 139 is prime.

%d bloggers like this: