Ideas in Geometry/Instructive examples/Lesson twenty Eight: Fields

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Lesson Twenty Eight: Fields
Section 4.3

We know a set of numbers can include any type of digit. We know there are set of numbers that are related to each other for example, the set of rational numbers, the set of negative numbers, the set of positive integers, the set of all integers. Depending on how the elements of the set are related to each other, we may also be able to describe some sets as fields. Now, not all sets are fields, but all fields are sets. In order for a set to be a field, certain things must be true about the elements.

A field is a set of numbers with binary operations, addition and multiplication, that upholds to specific axioms. The more familiar fields include the field of real numbers, the field of fractions, and the field of constructable numbers.

For a set of numbers to be considered a field, four properties must be true for the numbers in the set:

1. For all pairs of numbers, we'll just say "x,y" in the field, their sum, so x+y, must also be in the field. In other words the field, F, is closed under addition.

2. For all numbers in the field, "x", there is an additive inverse, "-x", in the field. So, x is any element of F, x + (-x) = 0 and (-x) is an element of F. In other words, any number in the field, it's negative is also in the field.

3. For all pairs of numbers, "x,y", in the field, the product of the numbers, "x * y", is also in the field. In other words, F is closed under multiplication.

4. For every number in the field, "x", there exists a multiplicative inverse, x-1, such that x-1 * x = 1 and x-1 is an element of F.

When determining if a set of numbers is a field, simply test the elements of the set for the axioms listed above.

Example: Is the set of natural numbers a field?

N= {0,1,2,3,4,...}

Test the first axiom: Is the set of natural numbers closed under addition? yes.

0+1=1 and 1 is an element of N also, 2+3=5 and 5 is an element of N

The second axiom: Is the additive inverse of each element also in the set of natural numbers? no.

2 is an element of N, however -2 is not an element and therefore set N is not a field

    • Once one of the properties does not hold true, we may conclude the set is not a field

Example: Fractions

Q= {-1,...-1/2,...,-1/4,...,0,...,1/4,...,1/2,....,1,...}

All numbers can be written as a fraction, for example 3= 3/1.

Test axiom one: Is set Q closed under addition? yes.

1/2 + 1/4 = 3/4 and 3/4 is an element of Q. also, 3/1 + 5/3 = 14/3 and 14/3 is an element of Q.

Test axiom two: Does each element of Q have an additive inverse in the set? yes.

-1/2 + 1/2 = 0 and both -1/2 and 1/2 are in set Q. also, -3 + 3 = 0 and both -3 and 3 are in set Q.

Test axiom three: Is set Q closed under multiplication? yes.

1/2 * 1/3 = 1/6 and 1/6 is also in set Q. Each number can be written as a fraction, and because both negative and positive fractions are included in set Q, all products will also be elements of Q.

Test the fourth axiom: Is there a multiplicative inverse for every element of Q? yes.

4/5 * 5/4 = 1 and both 5/4 and 4/5 are in set Q. All fractions are included, therefore the multiplicative inverses will always exist.