**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, bbe integers withb > 0.Then there exist unique integersqandrsuch that

*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:

Letaandbbe integers with . We say thatbdividesa(or thatbis a divisor ofa, or thatbis a factor ofa) if for some integer c. In symbols, “bdividesa” is written and “bdoes not dividea” is written (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

Letaandbbe integers, not both 0, and letdbe their greatest common divisor. Then there exists (not necessarily unique) integersuandvsuch that (page 12).

Hungerford’s Corollary 1.3

Letaandbbe integers, not both 0, and letdbe a positive integer. Thendis the greatest common divisor ofaandbif and only ifdsatisfies these conditions:

(i) and

(ii) if and , then (page 13)

Hungerford’s Theorem 1.4

If and , then .

*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 but Hungerford corrects this by saying that this is not implied. The one example provided (page 12) is . 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 . For this to be true let m=7 and n=-9. Exactly how 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 integerpispprimeif≠ 0, ±1 and the only divisors ofpare ±1 and ±p(page 16).

Hungerford’s Theorem 1.5

Letpbe an integer withp≠ 0, ±1. Thenpis prime if and only ifphas this property: whenever , then or (page 17).

Hungerford’s Corollary 1.6

Ifpis prime and , thenpdivides at least one of the (page 17).

Hungerford’s Theorem 1.7

Every integernexcept 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

Letn> 1, ifnhas no positive prime factor less than or equal to√n, thennis 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.