I'm just learning about sets, groups and fields, and I'm not sure how I'd go about making the addition cayley table for a field with, say, 4 or 5 elements. I know that, for example, 0 plus any element is the element itself, so for example:
But then I'm stuck. Assuming I don't know things like Lagrange's theorem (not about to prove that for a homework when I haven't even learnt about cosets n stuff). I've also heard about things like Galois field and how they're applicable to stuff like this, but I only know the very basics, and so I'm not sure what the next step would be. For example, why couldn't there be two different cayley tables for a field with 4 element:
and
Which one is violating a field axiom? I'm a bit unclear on this.
Thank you very much!
If one operation only is defined in your set, then you are in the realm of groups. To build up the Cayley table of a (finite...) group, you need to know how the operation "acts" on any pair ($2$-tuple) of elements of the set. Try with the set of the integers modulo $4$ with the operation $+$ (mod $4$), and then with set of the Cartesian product of two copies of the integers modulo $2$ with the addition component-wise (mod $2$).