0
$\begingroup$

I know that $\mathbb Z[\sqrt{-5}] \cong \mathbb Z[x]/(x^2 + 5)$, but how do I deal with quotients? For example, I've seen that

$$\mathbb Z[\sqrt{-5}]/(2, 1 + \sqrt{-5}) \cong \mathbb Z[x]/(x^2 + 5, 2, 1 + x)$$

but no matter how many other questions I look at, I can't find an answer that explains this rigorously, and in detail.

Another example is $\mathbb Z[x]/(2, 1 + x) \cong \mathbb Z/2\mathbb Z$. Again, I know that $\mathbb Z[x]/(2) \cong \mathbb Z_2[x]$ and $\mathbb Z[x] / (1 + x) \cong \mathbb Z$, and I also know by 3rd Isom. Thm that $\mathbb Z[x]/(2, 1 + x) \cong (\mathbb Z[x]/(2))/((2, 1+x)/(2))$, but I what is a rigorous explanation for why the result follows?

  • 0
    @Watson I've read that but many of the steps are not clear to me – 2017-02-08
  • 0
    Then explain clearly which steps are not clear to you. (Like 1) …… 2) ……… 3) ……). It will be more easier to answer. – 2017-02-08
  • 1
    You (the OP) should know questions like this one are more than likely to be closed. Your last question before this, which was identical, got two answers. It is not that likely you'll get a **very detailed and rigourous** explain beyond that (although there is perhaps people who love to write a lot...), so you likely won't be able to avoid a tutor, getting books and/or links to understand this more in depth. – 2017-02-08

0 Answers 0