Proving a structure is a field? Please help with what I am doing wrong here. It has been awhile since Ive been in school and need some help. The question is:
Let $F$ be a field and let $G=F\times F$. Define operations of addition and multiplication on $G$ by setting $(a,b)+(c,d)=(a+c,b+d)$ and $(a,b)*(c,d)=(ac,db)$. Do these operations define the structure of a field on $G$?
In order to be a field, the following conditions must apply:


*

*Associativity of addition and multiplication

*commutativity of addition and mulitplication

*distributivity of multiplication over addition

*existence of identy elements for addition and multiplication

*existence of additive inverses

*existence of multiplicative inverse 0 cannot equala, a-1*a=1


I started with 1. saying
$(a,b)+(c,d)+(e,f)=(a+c+e,b+d+f)$
$(a,b)+[(c,d)+(e,f)]=[(a,b)+(c,d)]+(e,f)$
$(a,b)+(c+e,d+f)=(a+c,b+d)+(e,f)$
$(a+c+d,b+e+f)=(a+c+e,b+d+f)$
which is not correct but I'm not sure where I went wrong. Is my logic incorrect?

Hint $\ (1,1) = (a,b)(a',b') = (aa',bb')\iff aa' = 1 = bb'.\,$ Therefore $\,(a,b)\,$ is invertible iff $\,a,b\,$ are. Employing this observation, it should be clear how to construct a noninvertible $\,(a,b)\ne (0,0)$ 
