set of symbols form a field

Hi all, I am going through a Algebra textbook and I need a bit of help with this question, so thank you in advance for any help.

Question: Prove that the set of symbols forms a field with nine elements, if the laws of composition are made to mimic addition and multiplication of complex numbers. Will the same method work for ? For ? Explain.

So I can see that the nine elements are: but I am confused as to where to go from here.

June

Re: set of symbols form a field

Well, write all the conditions for the set of symbols to form a field. For example, taking into account that with prime is a field, then. given then,

etc, etc,... . Show your work and we check it.

Re: set of symbols form a field

Thanks FernandoRevilla, sorry for my late reply, been gone a few days.

So,

Addition and Multiplication:

and

Inverse:

Distributive:

The complex addition identity is 0.

The complex multiplication identity is 1.

And commutative and associative follow easily.

Is this correct working out?

Thanks

Re: set of symbols form a field

And what about and , will the same method work for these?

Thanks

Re: set of symbols form a field

So I need to look at and to see if they form a field, if the laws of composition are made to mimic addition and multiplication of complex numbers.

So for the elements are:

So where do I go from here?

Any help guys?

Thanks

Re: set of symbols form a field