I have a set of numbers, how do I go about proving they form a field

andrey21

I have a set of numbers, how do I go about proving they form a field






Heres what I know

It has to be commutative under addition, which should give symmetry down the leading diagonal, which it does. What else must I show??

Thanks in advance

HallsofIvy

I don't quite understand what you are talking about. A "set of numbers"? That doesn't form a field- a single set is not a field. A field is a set of objects (which might or might not be numbers) together with two binary operations called "+" and "*". They must satisfy several laws:
1) They form a commutative group with addition (so, yes, commutative under addition but also associative, there exist an additive identity (0), and every member has an additive inverse.
2) Multiplication is commutative and associative and the distributive law holds. There is a multiplicative identity land every element except 0 (the additive identity) has a multiplicative inverse.

I have no idea what you mean by "symmetry down the leading diagonal". Are you referring to a diagonal in the additive or multiplcative tables? If so, that is just saying "commutative" again.

andrey21

Yes sorry I didn't word the question very well. I do have two tables for "+" and "*". In the + table is does have symmetry down the leading diagonal, so that is commutative. There does exist a zero, what do you mean by additive inverse?