To show that the three classical construction problems
are indeed impossible to solve we must use the theories of Groups, Rings
and Fields. We shall start with the definition of a group...
A Group is a set, G, along with a binary
operation, +, with the following four properties:
| (i) |
There exists an element, e, in G such that e+a
= a+e = a for any element a in G |
| (ii) |
For any a in G there exists an element b in G 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.
If a and b are in G then a+b
is also in G |
| (iv) |
The operation is associative, i.e.
a + (b + c) = (a + b) + c = a
+ b + c |
Some examples of Groups. |
...which we will then use to define a 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. If 1 exists
the R is called a ring with identity.
If R is commutative under * it is called a commutative
ring.
Some examples of Rings. |
Now we shall need to prove some basic theorems for Rings.
While some of these theorems may seem almost trivial they do serve to illustrate
an important point. Namely that the properties of rings must all
stem from the starting definition. While the use of the symbols +
and * for the two ring operations are intended to parallel our common definitions
of addition and multiplication (and guide our intutions to that effect)
we should be careful not to make any assumptions about their properties.
They are arbitrary operations acting on elements of an arbitrary set.
| Theorem 01:
Given a ring R then 0*x = x*0 = 0 for any x in R.
Proof
| 0*x = (0+0)*x |
by definition of the additive identity
(0) |
| 0*x = 0*x +
0*x |
by the distributive property |
| 0*x -(0*x) = 0*x + 0*x - (0*x) |
since the additive inverse always
exists in a Ring |
| 0 = 0*a |
by the definition of the additive inverse |
The proof for a*0 follows analogously. |
| Theorem 02:
Given x and y in a ring R then x*(-y) = (-x)*y = -(x*y).
Proof
| (-x)*y + x*y = (-x + x)* y |
by the distributive property |
| (-x)*y + x*y = (0)*y |
by the definition of the additive inverse |
| (-x)*y + x*y = 0 |
by Theorem 1 |
| (-x)*y = - (x*y) |
by the definition of the additive inverse |
The proof for x*(-y) follows analogously. |
| Theorem 03:
Given x and y in a ring R then (-x)*(-y) = (x*y).
Proof
The proof for x*(-y) follows analogously. |
Definition
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. |
While this |