Question

(c) Show that $T/l$consists of exactly P distinct co-sets

Short answer

$\left\{\left.I+r\right|0⩽r<p\right\}$are the consists of distinct co-sets.

Step-by-step solution

Determine T/lconsists of exactly two distinct co-sets

P is the co-sets of $T/l$ are $\left\{\left.I+r\right|0⩽r<p\right\}$

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

Assume that, $\frac{a}{b}\in T$ so that $a=pK+r$ and$b=pl+s$ for some $k,l,r,s\in \mathrm{\mathbb{Z}}$ with $0⩽r<p$ and $0<s<p$.

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

Use division algorithm

Using division algorithm,

There are $u,v\in \mathrm{\mathbb{Z}}$so that $rt=up+v$with$0⩽v<p$ then,

As $\left.p\right|-rtb+r+upb$then,

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

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