Question

(c) Show that $\mathit{T}\mathbf{/}\mathit{I}$consists of exactly$\mathit{p}$ distinct co-sets.

Short answer

$\left\{1+r/0\le r<p\right\}$are the consists of distinct co-sets.

Step-by-step solution

Determine T/I consists of exactly two distinct co-sets

$p$is the co-sets of$T/I$ are$\left\{1+r/0\le r<p\right\}$

These co-sets are distinct as for$0\le r,r\text{'}<p$ if $r\ne r\text{'}$then$p/r\text{'}-r$ so $r\text{'}-r\notin I$.

Assume that, $\frac{a}{b}\in T$so that $a=pK+r$and $b=pI+s$for some$k,I,r,s\in \square$ with$0\le r<p$ and $0<s<p$.

As$p$ is prime${\square }_p$ is field and there is$\left[t\right]\in {\square }_p$ so that $\left[t\right]\left[s\right]\mathbf{=}\left[1\right]$.

Use division algorithm

Using division algorithm,

There are$u,v\in \square$ so that$rt=up+v$ with$0\le v<p$ then,

As$p/-rtb+r+upb$ then,

$\frac{a}{b}=\frac{pk-rtb+r+rtb}{b}=\frac{pk-rtb+r+upb}{b}+v\in I+v$

Therefore,$T/I$ consists of exactly$p$ distinct co-sets.