Let $m$ be an integer, $m \geq 2$. and let ${\mathbb Z}_m$ be the set of all positive integers less than $m$, ${\mathbb Z}_m = \{0, 1, ..., m -1\}$. If $a$ and $b$ are in ${\mathbb Z}_m$, let $a + b$ be the least positive remainder obtained by dividing the (ordinary) sum of $a$ and $b$ by $m$, and, similarly, let $ab$ be the least positive remainder obtained by dividing the (ordinary) product of $a$ and $b$ by $m$. (Example: if $m = 12$, then $3 + 11 = 2$ and $3.11 = 9$.)
a) What is $-1$ in ${\mathbb Z}_5$?
b) What is $1/3$ in ${\mathbb Z}_7$?
I am unsure how $-1$ and $1/3$ can exist in the sets as it is meant to be made of positive integers and i know it forms a field when $m$ is a prime number, though i have n't been able to prove that yet either as i could not find justification for the following field properties.
- A(4) To every a there corresponds a unique scalar $-a$ such that $a + (-a) = 0$.
- B(2) Multiplication is associative $a(bg) =(ab)g$.
- B(4) To every non-zero scalar a there corresponds a unique scalar $a^{-1}$ (or $1/a$) such that $aa^{-1} = 1$.
Any help even just with the original two questions i think may give me enough to have another attempt at this proof.
Any help would be highly appreciated. Cheers