Chapter 8.1, Problem 21E

Describe the kernel of epimorphism $\varphi$ in Exercise $20$.

Consider the mapping $\varphi :Z\left[x\right]\to Z_k\left[x\right]$ defined by

$\varphi \left(a_0+a_{1}x+\cdot \cdot \cdot +a_n{x}^n\right)=\left[a_0\right]+\left[a_1\right]x+\cdot \cdot \cdot +\left[a_n\right]x^n$,

where $\left[a_i\right]$ denotes the congruence class of $Z_k$ that contains $a_i$. Prove that $\varphi$ is an epimorphism from $Z\left[x\right]$ to $Z_k\left[x\right]$.