1
$\begingroup$

Here is some background on adjoining elements from Artin:

Blockquote

I'm not sure I understand the "dropping bars" part. Why can we drop the bars?

Here is a theorem following this discussion:

enter image description here

  • 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 $\mathbb Z[\sqrt[3] 2]$ 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 $\mathbb Z[\sqrt[3] 2]$".
$\endgroup$
1
$\begingroup$

First picture: This is just some notational laziness/efficiency. You're correct that technically we should define $\overline{f} \in (R[x]/(f))[y]$ as $\overline{a_n} y^n + \cdots + \overline{a_0}$, and say $\overline{f}(\alpha) = 0$, but this is tiresome to write out. Moreover, in nice situations, for instance if $R$ is a domain, then $\pi: R[x] \to R[x]/(f)$ maps $R$ isomorphically onto its image if $f$ is nonconstant. In many applications $R = F$ is a field, and once we form $K = F[x]/(f)$ we identify $F$ with its image under the quotient map so that we can say that $F$ is a subfield of $K$. For instance, if we define $\mathbb{C} = \mathbb{R}[x]/(x^2+1)$, then $\mathbb{R}$ isn't literally a subfield (or even a subset) of $\mathbb{C}$, but is isomorphic to the subfield $\{\overline{r} : r \in \mathbb{R}\}$.

Second picture:

(i) Yes, this is the same "dropping bars" game. We write $a_i$ for $\overline{a_i}$ and just remember that we are working in the quotient.

(ii) He's basically just using the fact that remainders are unique in the division algorithm. If $\beta = r_1(\alpha)$ and $\beta = r_2(\alpha)$, where $r_1$ and $r_2$ both have degree $< n$, then $0 = r_1(\alpha) - r_2(\alpha)$, so $r_1(x) - r_2(x) \in (f)$. But if $r_1(x) - r_2(x)$ is nonzero then it has degree $< n$, contrary to the lemma. Thus $r_1 = r_2$ and the expression for $\beta$ is unique.

(iii) Sure, that is how the notation was defined in the proposition. I think here Artin is considering $\mathbb{Z}[\sqrt[3]{2}]$ as the smallest subring of $\mathbb{C}$ containing $\mathbb{Z}$ and $\sqrt[3]{2}$, as you mentioned above. If I recall correctly, earlier in the book Artin defines $$ \mathbb{Z}[\sqrt[3]{2}] = \{a + b \sqrt[3]{2} + c \sqrt[3]{2}^2 : a, b, c \in \mathbb{Z}\} $$ so that is another possible interpretation.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.