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Show that the elements 2,3, and $1 \pm \sqrt{-5}$ are irreducible elements of $\mathbb{Z}[\sqrt{-5}]$. I have never seen this notation before. From another post I am interpreting this to mean the following:

$\mathbb{Z}[\sqrt{-5}] = \{a_{0}+a_{1}\sqrt{-5}+ \dots + a_{n}(\sqrt{-5})^{n} \colon a_{i} \in \mathbb{Z} \}$.

Am I correct or is it something else? I do not need the proof, just verification of notation.

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  • $\begingroup$ Note that $\sqrt{-5}^2 = -5$, so you don't need all those terms. Also, the power should not be under the root - that will cause confusion on the future. $\endgroup$ – Thomas Andrews Feb 1 '16 at 16:37
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You are correct, but it it suffices to see that $\mathbb Z [\sqrt{-5}] = \{a_0 + a_1\sqrt{-5}: a_i \in \mathbb Z \}$

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    $\begingroup$ Why are the other powers of $\sqrt{-5}$ not needed? $\endgroup$ – Jack Feb 1 '16 at 16:38
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    $\begingroup$ Note that $(\sqrt{-5})^{2k} \in \mathbb Z$ and $(\sqrt{-5})^{2k+1} \in \sqrt{-5} \mathbb Z$ for every $k$ $\endgroup$ – flawr Feb 1 '16 at 16:41
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It's somehow correct, but it is misleading. The correct way to understand this notation is $$\mathbb Z[\sqrt{-5}]=\{a+b\mathrm i\sqrt5\,\mid a,b\in\mathbb Z\}$$

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    $\begingroup$ There's no reason to write $\sqrt{-5}$ as $i\sqrt{5}$... $\endgroup$ – paul garrett Feb 1 '16 at 16:47
  • $\begingroup$ @paulgarrett. The OP meant $\mathbb Z[\alpha]$ such that $\alpha^2=-5$. The notation $\sqrt{-5}$ is obsolete and misleading since $\sqrt{-5}\sqrt{-5}\neq\sqrt{(-5)(-5)}$. $\endgroup$ – Tom-Tom Feb 1 '16 at 19:50
  • $\begingroup$ No, it is certainly not obsolete at all. (The possibility of mis-using products of square roots is irrelevant, and bringing $\sqrt{-1}$ into the picture doesn't help anything.) $\endgroup$ – paul garrett Feb 1 '16 at 20:20
  • $\begingroup$ "bringing $\sqrt{-1}$ into the picture doesn't help anything". I can only agree with that, since this notation makes no more sense than $\log(-1)/37\pi$ or $0/0$. $\endgroup$ – Tom-Tom Feb 1 '16 at 21:03
  • $\begingroup$ I guess you and I are in different worlds of mathematical usage. I agree that square roots of negative numbers oughtn't be introduced to early... but in introductory abstract algebra or introductory algebraic number theory, writing $\sqrt{-5}$ or $\sqrt{-1}$ is certainly conventional usage. Not for grade-schoolers, no. Also, $\log(-1)$ is $\pi i$ ambiguous by multiples of $2\pi i$. Care is required, but it has sense. I think at the level of the question, $\mathbb Z[\sqrt{-5}]$ is apt, and is at least completely standard. $\endgroup$ – paul garrett Feb 1 '16 at 23:40

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