Let F be a field of which alpha, beta are components. Is it possible for (alpha)x(beta)=0, if alpha and beta are both nonzero?
Any ideas? Thanks for your help.
Really? Ok. First note that every field, by definition, is a commutative division ring. And thus, $\displaystyle \alpha\ne 0\implies \alpha^{-1}$ exists. So $\displaystyle \alpha\cdot \beta=0\implies \beta=0\cdot\alpha^{-1}$. But, it is relatively easy to see that $\displaystyle \alpha^{-1}\cdot 0=\alpha^{-1}\left(0+0\right)=\alpha^{-1}\cdot 0+\alpha^{-1}\cdot 0\implies \alpha^{-1}\cdot 0=0$ and thus $\displaystyle \beta=0$.