intro abstract algera - p.garret

Remark: If we already have the prime factorizations of two numbers m;n, then we can easily nd thegreatest common divisor and least common multiple.. Since this is true for eachprime, we

Intro Abstract Algebra

c1997-8, Paul Garrett, garrett@math.umn.eduhttp://www.math.umn.edu/~garrett/

(1) Basic Algebra of Polynomials

(2) Induction and the Well-ordering Principle

(8) Sun Ze's Theorem

(9) Good algorithm for exponentiation

(10) Fermat's Little Theorem

(11) Euler's Theorem, Primitive Roots, Exponents, Roots(12) (*) Public-Key Ciphers

(13) (*) Pseudoprimes and Primality Tests

(14) Vectors and matrices

(15) Motions in two and three dimensions

(16) Permutations and Symmetric Groups

(17) Groups: Lagrange's Theorem, Euler's Theorem(18) Rings and Fields: de nitions and rst examples(19) Cyclotomic polynomials

(27) Systems of Linear Congruences

(28) Abstract Sun Ze Theorem

(29) (*) The Hamiltonian Quaternions

(30) More about rings

(31) Tables

1 Basic Algebra of Polynomials

Completing the squareto solve a quadratic equation is perhaps the rst really good trick in tary algebra It depends upon appreciating the form of the square of the binomialx+y:

ax2+bx+c= 0can be rewritten as

from which the usualQuadratic Formulais easily obtained

For positive integersn, we have the factorialfunction de ned:

n! = 1
2
3
:::
(n,2)
(n,1)
n

Also, we take 0! = 1 The fundamental property is that

(n+ 1)! = (n+ 1)
n!And there is the separate de nition that 0! = 1 The latter convention has the virtue that it works out inpractice, in the patterns in which factorials are most often used

Thebinomial coecientsare numbers with a special notation



nk

#1.1Factorx6 ,y6 in two dierent ways.

#1.2While we mostly know thatx2 ,y2has a factorization, thatx3 ,y3 has a factorization, thatx3+y3

has, and so on, there is a factorization that seldom appears in `high school': x4+4y4has a factorization intotwo quadratic pieces, each with 3 terms! Find this factorization Hint:

x4+ 4y4= (x4+ 4x2y2+ 4y4),4x2y2= (x2+ 2y2)2 ,(2xy)2

2 Induction and the Well-ordering Principle

The meaning of the word `induction' within mathematics is very dierent from the colloquial sense!First, letP(n) be a statement involving the integern, which may be true or false That is, at this point

we have a grammatically correct sentence, but are making no general claims about whether the sentence istrue, true for one particular value ofn, true for all values ofn, or anything It's just a sentence

Now we introduce some notation that is entirely compatible with our notion of function, even if thepresent usage is a little surprising If the sentenceP(n) is true of a particular integern, write

P(n) = trueand if the sentence asserts a false thing for a particularn, write

P(n) = falseThat is, we can viewP as a function, but instead of producing numbers as output it produces either `true'

or `false' as values Such functions are calledboolean

This style of writing, even if it is not what you already knew or learned, is entirely parallel to ordinaryEnglish, is parallel to programming language usage, and has many other virtues

Caution: There is an another, older tradition of notation in mathematics which is somewhat dierent,which is and which is harder to read and write unless you know the trick, since it is not like ordinary English

at all In that other tradition, to write `P(n)' is to assert that the sentence `P(n)' is true In the othertradition, to say that the sentence is false you write `:P(n)' or `P(n)'

So, yes, these two ways of writing are not compatible with each other Too bad We need to make achoice, though, and while I once would have chosen what I call the `older' tradition, now I like the rst waybetter, for several reasons In any case, you should be alert to the possibility that other people may chooseone or the other of these writing styles, and you have to gure it out from context!

Principle of Induction

 IfP(1) = true, and

 ifP(n) = true impliesP(n+ 1) = true for every positive integern,

 thenP(n) = true for every positive integern

Caution: The second condition does not directly assert that P(n) = true, nor does it directly assertthatP(n+ 1) = true Rather, it only asserts a relative thing That is, more generally, with some sentences

AandB (involvingnor not), an assertion of the sort

(AimpliesB) = truedoes notassert thatA= true nor thatB = true, but rather can be re-written as conditional assertion

if (A= true) thenB= true

In other words we prove that an implication is true

That is, pushing this notation style a little further, we usually prove

(A= true) impliesB= true) = true

In the more traditional notation, the assertion of Mathematical Induction is

 IfP(1), and

 ifP(n) impliesP(n+ 1) for every positive integern,

 thenP(n) for every positive integern Even though I am accustomed to this style of writing, in the end

I think it is less clear!

Another Caution: Whatever the notation we use, the statements above do not indicate the way that

we usually go about proving something by induction Rather, what we use is Practical Paraphrase of

`Principle of Induction':

 First, proveP(1) = true

 Second, assumeP(n) = true and using this proveP(n+ 1) = true (for every positive integern)

 Then conclude P(n) = true for every positive integern

The second item in this procedure is what is usually calledthe induction step Our paraphrase makes

it look a little dierent than the more ocial version: in the ocial version, it looks like we have to provethat an implication is correct, whereas by contrast in our modi ed version we instead assume something trueand see if we can then prove something else

The most popular traditional example is to prove by induction that

1 + 2 + 3 + 4 +:::+ (n,2) + (n,1) +n= 12n(n+ 1)Let P(n) = true be the assertion that this formula holds for a particular integer n So the assertion

P(1) = true is just the assertion that

1 = 121(1+1)which is indeed true To do the induction step, we assume that

1 + 2 + 3 + 4 +:::+ (n,2) + (n,1) +n+ (n+ 1) = 12n(n+ 1) + (n+ 1)

The left-hand side is just what we want, but the right hand side is not But we hope that it secretly is what

we want; that is, we hope that

1

2n(n+ 1) + (n+ 1) = 12(n+ 1)((n+ 1) + 1)

We have to check that this is true

This raises an auxiliary question, which is easy enough to answer once we make it explicit: how would

a person go about proving that two polynomials are equal? The answer is that both of them should besimpli ed and rearranged in descending (or ascending) powers of the variable, and then check that corre-sponding coecients are equal (And this description de ... dierent proof concept works better:

Well-Ordering PrincipleEvery non-empty subset of the positive integers has a least element.This Well-Ordering Principle sounds completely innocuous,...

The left-hand side is just what we want, but the right hand side is not But we hope that it secretly...

2 Induction and the Well-ordering Principle

The meaning of the word `induction'' within mathematics