Algebraic Structures — Field

Mathematics does not deal so much in numbers, but in “mathematical structures.”  These are basically abstract concepts that fit together according to certain rules or patterns.

One abstract concept that is very important is that called a Set.  A set in mathematics is a very abstract concept, but unless we are involved in higher mathematics, a set is something very similar to what we call a “collection” in everyday English.  Since we are not involved in higher math in this course, we will consider a set to be a collection of some objects (often numbers, but it could be a collection of other things, such as functions, triangles, etc.).  Many books will define a set to be a collection.  This definition is not correct, so as a matter of honesty, we point to that it is not correct.  However, the differences between a set and a collection are so subtle and abstract, we need get involved, but will consider a set to be a collection.

Another important concept is that of an operation—it means a process where some members of a set are combined together to get something new, like mixing flour and water to get dough.  In mathematics, in dealing with numbers, the most common operations are addition, subtraction, multiplication, and division.  There are others, of course, but we need not be concerned with them.  In dealing with functions, the most important operations are there are these four and a FIFTH operation that is important, called composition.  We studied it in class–f◦g means f(g(x)).

An operation is called binary, if, by definition we must always combine exactly two things together.  All the above operations are binary.  Note we cannot combine three numbers with addition together, for example, we cannot add 2+3+4 all at once.  We must first add only two numbers together, get a temporary answer, and then we add the third number to this temporary answer.  This means the operation is binary—we can only deal with two at a time, although we can keep adding more numbers to our answer after we get it.

An operation is called unique, if, unlike tossing a pair of dice, for example, combining the same two inputs always produces the same output.

Rules for addition:

1. The sum of any two real numbers a and b is a real number written a+b

This is called the Law of Closure for addition of Real numbers.

2. For all real numbers a, b, c:  (a+b)+c=a+(b+c)

This is called the Associative Law for addition of Real numbers.

3. There is a one unique real number called zero, written 0, such that a+0 = a and 0+a = a  for all real a.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems.)   That is to say, that there is one and only one real number that can be used for I to make the following two equations come true:  For any Real number, a, a+I = a and I+a = a  for all real a.  (And of course, I is 0).

This is called the Additive Identity Law for addition of Real numbers.

4. For each real number a there is a real number called –a such that a+(-a)=0 and

    (-a)+a = 0.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems.)

This is called the Inverse Law for addition of Real numbers.

5. a+b=b+a

This is called the Commutative Law for the addition of Real numbers.

Rules for multiplication:

1. The product of any two real numbers is a real number written ab or ab

2. For all real numbers a, b, c:  (ab) c=a (bc)

3. There is a real number called one, written 1, such that a1=a and 1a=a for all real a.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems).

4. For each real number a0,  there is a real number called 1/a such that
     a (1/a)=1 and (1/a) a= 1.   1/a is also written as a-1.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems.)

5. ab=ba

Rule 11 connecting addition and multiplication:

a(b+c)=ab+ac

Rule 1 is called the Law of Closure.

Rule 2 is called the Associative Law.

Rule 3 is called the Identity Law.

Rule 4 is called the Inverse Law.

Rule 5 is called the Commutative Law.

Rule 11 is called the Distributive Law.  (Adding columns and rows—switching order.)

Note:

1.  Rule 2 does not make sense unless Rule 1 is first established.

2.  Rule 4 does not make sense unless Rule 3 is first established.

3.  If a system has just one binary operation and the first four laws, it is called a “Group.”  Since Rule 5 is not necessarily true in an arbitrary group, it is necessary to state the identity and inverse laws on both sides.

4.  If a system has just one binary operation and the first five laws, it is called a Commutative Group or an Abelian Group.

5.  If a system has rule 2 and 5, then the General Associative Law is true, which says you can change the order and the parentheses any way you want.

6.  If a system has two binary operations and all 11 laws, then it is called a field.

You should be able to state these rules in the abstract—If S is a set, and if a, b, and c are any arbitrary members of S, and if * is a unique binary operation defined for the members of S, then (commutative law, for example, would be a*b = b*a.

Now some results or consequences of these rules:

1.  If a+c=a+b then b=c.  (This will be true in any group that has + as an operation.)

2.  If ab=ac, then b=c, providing a0  (This will be true in any group that has multiplication as an operation.)

3.  These laws enable us, if we are in a field,  to solve any equation of the following forms:

a+ x=b,

x+a=b,

ax=b (provided a0),

xa=b (provided a0), and

ax + b = c

4.  a0=0  for any Real number a.

5.  ab = 0 means a = 0 or b = o or both.

6.  The zero principle:  In a field, ab = 0if and only ifa = 0 or b = 0 or both.

This is a consequence of #4 and #5 above.  We needed all 11 laws (except for the commutative laws) to prove #3 and #4, so we see the zero principle will always hold in a field, but not necessarily in a group.

5 – 7

7.  (a)  b = (ab)  (Henceforth, we will omit the  for multiply, or use a dot instead.)

All the above represent the basic algebraic properties of a field.  There are many fields besides the real numbers.  Some are finite  and some are infinite, such as , , and –and there are many other examples of both finite and infinite fields not mentioned.  Any system that is a group will look a lot like our number system, , the real numbers.  However, any field will look even more like .

To characterize the Real Numbers mathematically, we need to say it is an infinite field with the following additional properties—these additional properties, however, are generally not considered “algebraic” properties, but “geometric” properties:

It is infinite, ordered, Archimedian, and complete.  These three properties are explained briefly below, but you are not responsible for them.

Ordered means given any two different numbers, then one is greater than the other (or equivalently, one is smaller than the other).

Archimedian means that given any small number, s, regardless of how small, and a big number, b, regardless of how big, then it is possible to add enough s’s together so that eventually the sum  s + s + … + s > b.  In other words there is a number n, so that ns > b.  For example, is s is 1/100 and b is 1,000,000 we can keep adding 1/100 + 1/100 etc. until eventually the sum is bigger than 1,000,000.  Equivalently, if n is any number larger than 100,000,000 then ns > b.