Question

Let F be a field and $\mathit{p}\left(x\right)\mathbf{\in }\mathit{F}\left[x\right]$. Prove that ${\mathit{F}}^{\mathbf{*}}\mathbf{=}\left\{\left[a\right]a\in F\right\}$is a subring of $\mathit{F}\mathbf{\left[}\mathit{x}\mathbf{\right]}\mathbf{/}\mathbf{\left(}\mathit{p}\mathbf{\right(}\mathit{x}\mathbf{\left)}\mathbf{\right)}$.

Short answer

It is proved thatF* is subring by theorem 3.2.

Step-by-step solution

 Determine F* is a subring

Consider the given subset $F^{*}=\left\{\left[a\right]a\in F\right\}$.

The subset $F^{*}=\left\{\left[a\right]a\in F\right\}$satisfies the following conditions:

I. Addition property

If $\left[a\right],\left[b\right]\in F^{*}$, then $\left[a\right]+\left[b\right]=\left[a+b\right]\in F^{*}$; thus, F*is closed under addition.

ii. Multiplication property

If $\left[a\right],\left[b\right]\in F^{*}$, then $\left[a\right]·\left[b\right]=\left[ab\right]\in F^{*}$; thus, F* is closed under multiplication.

iii. $\left[0\right]\in F^{*}$

iv. For all $\left[a\right]\in F^{*},-\left[a\right]=\left[-a\right]\in F^{*}$; thus, F* is closed under additive inverse.

Therefore, F* is subring by theorem 3.2.