simple algebraic extensions with the same minimal polynomial

Question

I can't see why $ji^{-1}$ is the identity on $K$, could someone explain please

Answer 1

By definition:

$$ i:K[t]/\langle m\rangle\to K(\alpha)\;,\;\;i(t+\langle m\rangle):=\alpha$$

$$j:K[t]/\langle m\rangle\to K(\beta)\;,\;\;j(t+\langle m\rangle):=\beta$$

Observe that in both cases and $\;\forall\,k\in K\;,\;\;i(k+\langle m\rangle)=j(k+\langle m\rangle)=k\;$ , as $\;K\;$ is embedded in $\;K[t]/\langle m\rangle\;$ precisely in that way: $\;k\mapsto k+\langle m\rangle\;$

Thus, we have that

$$\forall\,k\in K\;,\;\;ji^{-1}(k)=j(i^{-1}(k))=j(k+\langle m\rangle)=k$$