Ring Theory as Applied to the Three Classical Geometric Construction Problems

 
Glossary of Symbols
[E:F]
  "degree of E over F"
dim(E)
  "dimension of the vector space E"
 F(S)
  "F adjoin S"
A  B
  "Union of A with B"
A  B
  "Intersection of A with B"
 "is an element of"
 "there exists"

 
Glossary of Terms
Adjoin Intersection
Algebraic Numbers Inverse
Commutative Ring Ring
Constructible Circle Ring with Identity
Constructible Line Rational Numbers
Constructible Numbers Real Numbers
Degree Subfield
Extension Subgroup
Field Transcendental Numbers
Group Union
Identity Unit (of a Ring)
Integers Vector Space
Adjoin
Let F be a field, E be an extension of F, and S be some subset of E.  Let H be the set of all subfields of E which contain both F and S.  We define F adjoin S to be the intersection of all elements (sets) in H and we write it as F(S).
Algebraic Numbers
The algebraic numbers are the set of all roots of finite polynomials.
Commutative Ring
 
A ring which is commutative under multiplication is called a Commutative Ring.
Constructible Circle
A constructible circle is a circle which has its center on a constructible point and has at least one constructible point lying on its circumference.  Algebraically it can be represented by an equation of the form:
x2+y2+ax+by+c = 0     
where a,b, and c are all constructible numbers.
Constructible Line
A constructible line is a line drawn through two constructible points.  Algebraically it can be represented by an equation of the form:
ax + by + c = 0 
where a,b,and c are all constructible numbers.
Constructible Numbers
 
The constructible numbers is the set of all numbers (lengths) which can be constructed using a straight-edge and compass in a finite number of steps.  It will be designated by the symbol C.  
Degree
 
Let E be an extension of F.  If the dimension of E as a vector space over F is finite then by the degree of E over F we mean the dimension of E as a vector space over F.  We will write [E:F] to mean the degree of E over F.

Examples:
[Q( root 2 ):Q] = 2
[Q( cube root2):Q] = 3
[Q( i ):Q] = 2      (where i is the imaginary number)
[Q( root 2, root3 ):Q] = 4
Extension
 
Let F be a subset of a field E, such that F is also a field.  Then E is said to be an extension of F. 
Field
A Field is a commutative ring in which all non-zero elements are units.  In other words you can talk about "dividing" in a Field.
Group
 
A Group is a set, G, along with a binary operation, +, with the following four properties:
 
(i) eG such that e+a = a+e = a aG
(ii) aG  bG such that a+b = b+a = e

b is called the inverse of a and is written b = -a
(iii) G is closed under the operation, i.e.

     a, b G  a+b G 
(iv) The operation is associative, i.e.

     a + (b + c) = (a + b) + c = a + b + c

Some examples of Groups. 
Identity
 
Let * be a binary operation.  If there exists an element e such that for any element x:   x*e = e*x = x  then e is called the identity for that binary operation. 

For the operation represented by * the identity is commonly written as 1. 

For the operation represented by + the identity is commonly written as 0.
Integers
 
The set of integers, Z, is the set of all natural numbers along with their additive inverses and zero.  e.g., 
Z = {...,-2,-1,0,1,2,...}
Intersection
 
The intersection of two sets A and B is a new set that contains all of the elements that are in both A and B.
Inverse
 
Let * be a binary operation and the identity with respect to * be 1.  If some element, x, has an inverse, y, (with respect to *) then x*y = 1 and y is written as x-1. 

If the binary operation is written as + and has identity 0, then the inverse of x is commonly written as -x.
Ring
 
A Ring is a non-empty set, R, on which there are defined two operations 
"+" and "*" such that
 
(i) (R, +) is an abelian (commutative) group
(ii) (R, *) is associative and closed
(iii) it has the following distributive properties:
a * (b + c) = a*b + a*c
(b + c) * a = b*a + c*a

Call the "+" operation addition and denote its identity by the symbol 0.

Call the "*" operation multiplication and if an identity for this operation exists denote it by the symbol 1.

Some examples of Rings.
Ring with Identity
 
A ring for which the multiplicative identity exists is called a ring with identity. 
The multiplicative identity is denoted by the symbol 1.
Rational Numbers
 
The set of rational numbers, Q, is the set of all integers, along with all of their possible quotients, e.g. 
Q = {(x/y) | where x and y are both integers, y not equal to 0}
Real Numbers
 
The real numbers are the set of all rational numbers plus the set of all numbers that can be defined as limits to a sequence of rational numbers.  e.g. it is the rational numbers along with  all numbers "in-between."
Subfield
 
Let F be a subset of a field E, such that F is also a field.  Then F is said to be an subfield of E. 
Subgroup
Let G be a subsets of a group H, such that G is also a group.  Then G is said to be a subgroup of H.
Transcendental Numbers
 
The transcendental numbers are the set of all numbers that are not algebraic.  Some examples of transcendental numbers are pi, and e (Napier's constant).
Union
The union of two sets A and B is a new set that contains all of the elements that are in either A or B.
Unit (of a Ring)
 
If x is an element of a ring R with identity and x has an inverse with respect to * then we call x a unit of R. 

We denote the set of all units in R by UR. 

Example:

Say R=Z (the integers) then UR={-1, 1}
Vector Space
A Vector Space of a field F is a non-empty set V together with two operations
+:  V + V V
*:  F * V V
Where the following properties hold:
  1. (V,+) is an abelian group
  2. for any a in F and x in V the product a*x is in V
  3. for any a in F and x, y in V the product a*(x+y) = a*x+a*y
  4. for any a,b in F and x in V the product (a+b)*x = a*x + b*x
  5. for any a,b in F and x in V the product (a*b)*x = a*(b*x)
  6. 1*x = x for any x in V
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