If I said $\forall h \neq 0$ $\exists$ t I know that this can be translated into for any given t there exists, but does t have to depend on h? I think not, but can someone tell me if I'm correct?
For instance, if I wanted to prove every finite integral domain is a field, I can say that I have a non-repeating set which contains all the elements of the integral domain ${a_1,a_2,a_3....,a_n}$ for some n, then multiply every element by a non-element and so ${ca_1,ca_2...,ca_n}$ is the same finite integral domain and since it is a ring with unity there exists a $1\in$ R, so c$a_i$ $=$ $1$ and so there exists an inverse, however this inverse does not necessarily depend on $a_m$. May someone please explain?
The definition of a field is a commutative ring such that $\forall a\neq$ 0 in R $\exists$ $a^{-1}$ in R such that $aa^{-1}$ $=$ $1$.