Here is some background on adjoining elements from Artin:
I'm not sure I understand the "dropping bars" part. Why can we drop the bars?
Here is a theorem following this discussion:
- Red part: Why do elements of $R[\alpha]$ have the form $g(\alpha)$? It is certainly clear if I use the definition of $R[\alpha]$ saying that this is the smallest ring containing $R$ and $\alpha$. But in this context $R[\alpha]$ is a name for the ring $R[x]/(f)$, and don't understand why elements of $R[x]/(f)$ are polynomials in $\alpha$. The elements of $R[x]/(f)$ are, in my understanding, polynomials of the form $\sum \overline a_i \overline x^i=\sum \overline a_i \alpha^i$, and again, if we can "drop bars", this is just $\sum a_i \alpha^i$, and my question is answered, but why can we do that?
- Blue part: How does uniqueness follow from this?
- Green part: I don't understand why Artin writes "isomorphic" here. Doesn't he define $\mathbb Z[\sqrt 2]$ to be $\mathbb Z [x]/(x^3-2)$? I'm also not sure why Artin refers to that substitution map and mentions the kernel instead of just saying "consider $\mathbb Z[\sqrt 2]$".