Surely you understand that is the set of rational numbers.
What can one say about the rationals with respect to the listed axioms?
Tell whether or not with the two operations ⊕ and (in fact I want to type the but with a circle as in ⊕ but I don't know how to) is a field.
Looking to my class notes, a field is a set with 2 operations " " and " " such that
1)
2)
3) element " " such that
4) , such that
5)
6)
7) element such that
8) , such that
9) .
So to solve my problem, I have to check if any and elements of respect the 9 rules. But I'm not able to start. How can I check if any element and of are such that ? I thought about writing down and But it doesn't solve the problem. (Note that and ).
I also see that a field is a set with 3 elements as minimum. (unless , an can be the same element.)
Ok, I'll try to do it.
Yes they are rational ( with and where ). I'll try to prove they are unique (I think it's not that hard by reductio ad absurdum).
But even then, I will just have almost proved part 3) and 7).
I'm not sure I understand well the problem. For part 1... Have I to show that ? Or just assume it? If I have to prove it, how can I do so?
direct proofs will suffice.
assume there are other elements with the same property and show that they have to be equal to the additive and multiplicative identity elements that you already defined.
you must show it. add the fractions on both sides as you would normally. of course, addition of fractions in is defined as:But even then, I will just have almost proved part 3) and 7).
I'm not sure I understand well the problem. For part 1... Have I to show that ? Or just assume it? If I have to prove it, how can I do so?
and we know that two fractions, say and are equivalent (by definition) if
Thanks Jhevon!
Ok for part 3 and 7, I'll try later. I prefer to do it following the order.
You said that so what I did in post 3 : is in fact wrong because in the numerator I have instead of ? I know that both are equal but I can't assume it since one property I have to show is precisely that ! That's really limitating me. Maybe I should learn precisely the definition of the set, so I can think faster about how things work.
yes, you have to prove commutativity in addition and multiplication first. it will make proving the harder claims easier when you have those basic properties under your belt
you might have to use it if you do my method as well. i'm too lazy to expand and see. it is the standard way though.