Mathematics Chronology

2:30 PM

I have read and reread "Fearless Symmetry" chap 8 Galois Groups a few times during the last week. Now to make a few notes.

The book is divided into three main parts:

• Part One. Algebraic Preliminaries (chaps. 1 - 7)
• Part Two. Galois Theory and Representations (chaps. 8 - 16)
• Part Three. Reciprocity Laws (chaps. 17 - 23)

I am thoroughly enjoying this book. It is quite different than other books I have read in the last couple of years. I would categorize these books as follows: books that describe how to "do" certain things (e.g. textbooks) books on the history of mathematics general books on mathematics, but no "real" mathematics books that focus on the ideas and concepts of mathematics (this book!)

"Fearless Symmetry" is largely about terminology (i.e. concepts) and notation.

Let me try to review the previous seven chapters that constitute Part One:

Chapter	Terminology (concepts)	Notation
Part One	Algebraic Preliminaries
1	Representations
	set
	one-to-one correspondence
	standard object
	function
	morphism
	representation
2	Groups
	group
	group law
	discrete groups
	integers under addition
	permutation group
	continuous groups
	Lie groups
	rigid motions in space
	rotations of a circle	SO(2)
	rotations of a sphere	SO(3)
	set of real numbers under addition	R
3	Permutations
	group of permutations
	identity permutation	e
	cycle decomposition
4	Modular Arithmetic
	conguence
	modulus
	integer
	prime number
	field
5	Complex Numbers
	real number
	complex number	a + bi
	complex conjugate	a - bi
	set of all complex numbers	C
	algebraic closure
6	Equations and Varieties
	rational number
	irrational number
	set of all rational numbers	Q
	Z-equation
	set of all integers	Z
	set of integers modulo p
	variety
	roots of a polynomial
7	Quadratic Reciprocity
	Legendre symbol
	Quadratic Recirocity	Suppose that p and q are odd primes. 1. 2. 3. If , then . 4. If p or q or both , then .
Part Two	Galois Theory and Representations
8	Galois Theory
	algebraic numbers
	set of all algebraic numbers
	algebraically closed field
	absolute Galois group	G

I am not happy with the above table (yet). Although it identifies many of the important concepts that have been discussed so far, it fails to clarify exactly what these concepts are. I need to add one more column that does this. I also need to add at least one example to illustrate the nature of the concept.

Chapter	Terminology (concepts)	Description	Notation
Part One	Algebraic Preliminaries
1	Representations
	set	A set is a collection of things, often called the elements of the set.	{a, b, c}
	one-to-one correspondence	A one-to-one correspondence from a set A to set B is a rule that associates to each element in A exactly one element in B, in such a way that each element in B gets used exactly once, and for exactly one element in A.
	standard object	This is an object that we know quite a bit about.
	function	A function from a set A to a set B is a rule that assigns to each element in A an element of B.
	morphism	A morphism is a function from A to B that captures at least part of the essential nature of the set A in its image in B.
	representation	A representation is a morphism (i.e. a function) from a source object to a standard target object. If A represents B, we have three things: two objects, A and B, which in this context will be sets, and the relation between them, which will be a morphism. When A and B have some additional "structure" (e.g. they are finite sets, or ordered sets, or ...) and we restrict the possible morphisms from A to B to have something to do with that structure.
	Example: Let A be the set {red, blue, yellow, pink}. Let B be the standard object {1, 2, 3, 4}. Let f be the following rule: red is associated with 1, blue is associated with 2, yellow is associated with 3 and pink is associated with 4. f is a morphism because because this rule is capturing something about the number of elements in A. Therefore we may consider f to be a representation of A.
2	Groups
	group	A group G is a set with a composition defined on pairs of elements such that : 1. There is a neutral element e in G, so that no matter what element in the group is substituted for x. e is sometimes called the identity element. 2. For any element x of G, there is some element y in G so that . That is, every element has an inverse element. 3. For any three elements, x, y, and z in G, we have . This is called the associativity of the composition. Each group has its own law of composition. It can be whatever we define it to be (e.g. addition, multiplication, rotation, ...).
	group law	If G is a group, the group law is the rule that tells how to combine two elements in the group to get the third.
	discrete groups	There is no smooth path from one element to another. (e.g. the integers)
	integers under addition
	permutation group
	continuous groups	There are infinitismal differences between elements.
	Lie groups	Pronounced "Lee", named after the Norwegian mathematician Sophus Lie who studied them.
	rigid motions in space
	rotations of a circle		SO(2)
	rotations of a sphere		SO(3)
	set of real numbers under addition		R
	Examples: Consider the set Z of all the integers (positive, negative and zero). Z is an infinite set. Let the group law be familiar addition. Then Z under this group law is a group. Consider the set of three rotations of 60, 120 and 180 degrees of an equilateral triangle. The rotations constitute the group law. The elements of the group are the three positions (i.e. orientations) of the triangle. Groups are used to describe various types of symmetries.
3	Permutations
	group of permutations
	identity permutation		e
	cycle decomposition
4	Modular Arithmetic
	conguence
	modulus
	integer
	prime number
	field
5	Complex Numbers
	real number
	complex number		a + bi
	complex conjugate		a - bi
	set of all complex numbers		C
	algebraic closure
6	Equations and Varieties
	rational number
	irrational number
	set of all rational numbers		Q
	Z-equation
	set of all integers		Z
	set of integers modulo p
	variety
	roots of a polynomial
7	Quadratic Reciprocity
	Legendre symbol
	Quadratic Recirocity		Suppose that p and q are odd primes. 1. 2. 3. If , then . 4. If p or q or both , then .
Part Two	Galois Theory and Representations
8	Galois Theory
	algebraic numbers
	set of all algebraic numbers
	algebraically closed field
	absolute Galois group		G