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
which is not correct but I'm not sure where I went wrong. Is my logic incorrect?