posted by Sherry on .
Tell whether or not the following sets are fields under the operations + and *. If the set is not a field, tell which one(s) of the Field Axioms do not apply:
a) [rational numbers]
c) [positive number]
d) non-negative numbers]
That is a question from my text. Can someone please explain this to me? My teacher said something about 44 examples (because there are 11 field axioms).
Eleven seems rather a lot: according to Wolfram MathWorld (I can't post the address, it seems, but just google "field axioms" to find it) there are five: Associativity, Commutativity, Distributivity, Identity and Inverses. Algebraic definitions of these are given there.
The same site supplies this information about fields: "A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra". Can you apply these to your four sets?