Question

Show that $\mathbf{2}{\mathbf{x}}^3+\mathbf{4}{\mathbf{x}}^{\mathbf{2}}+\mathbf{8}\mathbf{x}+\mathbf{3}\in {\mathrm{\mathbb{Z}}}_{\mathbf{16}}\left[x\right]$has no roots in any ring K that contains${\mathrm{\mathbb{Z}}}_{16}$ as a subring. [See Exercise 11.]

Short answer

It is proved that$2x^3+4x^2+8x+3\in {\mathrm{\mathbb{Z}}}_{16}\left[x\right]$ has no roots in any ring K, which contains${\mathrm{\mathbb{Z}}}_{16}$ as a subring.

Step-by-step solution

Statement of Exercise 3.2.17

Exercise 3.2.17states that if $u$isa unit in a ring$R$ with an identity, then $u$willnot be a zero divisor.

Show that 2x3+4x2+8x+3∈ℤ16x has no roots in any ring K that contains ℤ16 as a subring

If a ring$K$ contains${\mathrm{\mathbb{Z}}}_{16}$and$u$ is the root of $2x^3+4x^2+8x+3$, then:

role="math" localid="1649241238121" $\left(2u^3+4u^2+8u\right)·5=\left(-3\right)·5$

$$\begin{gathered} =-15 \\ =1 \end{gathered}$$

Therefore, 2 willbe the unit in $K$. However, it is observed that$2·8=0$ in $K$.

As a result, it implies that 2 is both a unit and a zero divisor, which contradictsExercise3.2.17.

Hence, it is proved that$2x^3+4x^2+8x+3\in {\mathrm{\mathbb{Z}}}_{16}\left[x\right]$ has no roots in any ring K that contains${\mathrm{\mathbb{Z}}}_{16}$ as a subring.