Question

Let $\mathit{J}$be the set of all polynomials with zero constant term in $\mathit{\mathbb{Z}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$.

(b) Show that $\mathit{\mathbb{Z}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{/}\mathit{J}$consists of an infinite number of distinct co-sets, one for

each $\mathit{n}\mathbf{\in }\mathit{\mathbb{Z}}$.

Short answer

It is proved$\mathbb{Z}\left[x\right]/J=\{n+J|n\in \mathbb{Z}\}$ consists of an infinite number of distinct co-sets.

Step-by-step solution

Consider the polynomial

It is given that $J$is the set of all polynomials with zero constant terms in $\mathbb{Z}\left[x\right]$.

Consider the polynomial${\sum }_{i=0}^k{a}_i{x}^i\in \mathbb{Z}\left[x\right]$.

Determine ℤ[x]/J.

The above polynomial can be written as follows:

$\left({\sum }_{i=1}^k{a}_i{x}^{i-1}\right)x+a_0\in a_0+J$

Similarly, for $m,n\in \mathbb{Z}$, if $m\ne n$, then

$x|m-n$. Thus, $m-n\notin J$and $m+J\ne n+J$

Therefore, $\mathbb{Z}\left[x\right]/J=\{n+J|n\in \mathbb{Z}\}$.

Hence, it is proved$\mathbb{Z}\left[x\right]/J=\{n+J|n\in \mathbb{Z}\}$ consists of an infinite number of distinct co-sets.