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If $\mathcal O=\mathbb Z\left[\frac{1+\sqrt{5}}{2}\right]$, I'd like to show that $\left(\mathcal O:\mathbb Z \left[\sqrt{5}\right]\right)=2$. It's to show that Dedekind's factorisation theorem doesn't apply to the prime 2. I feel like this should be some basic computation and I'm just over thinking it/forgetting some of my algebra, but what is the best way to do this? I tried to define a map from $\mathcal O$ to $\mathbb F_2$ with kernel $\mathbb Z\left[\sqrt{2}\right]$ but things weren't working out. I was also thinking of looking at these things as rank 2 $\mathbb Z$-modules, but this seemed like overkill maybe. Thanks.

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  • $\begingroup$ How do you define this index? $\endgroup$ – Bernard Feb 21 '19 at 19:42
  • $\begingroup$ @Bernard I think the OP wants to find the index of $\mathbb Z[\sqrt 5 ]$ as a subgroup of $\mathcal O$, i.e. the OP want to find the ideal norm. $\endgroup$ – Kenny Wong Feb 21 '19 at 19:45
  • $\begingroup$ Not sure if this is the first appearance. Take a look anyway. $\endgroup$ – Jyrki Lahtonen Feb 24 '19 at 5:05
  • $\begingroup$ This feels a bit better. And this seems to be specifically about rings of integers in a number field. $\endgroup$ – Jyrki Lahtonen Feb 24 '19 at 5:14
  • $\begingroup$ @JyrkiLahtonen on this material and my one-time mse blog, I just ordered this new book by an author I like (for ternary forms; the book will mostly relate binary forms and quadratic number fields) bookstore.ams.org/dol-52?_zs=JL6BH1&_zl=Llbu4 OR bookstore.ams.org/dol-52 author J. L. Lehman, book Quadratic Number Theory, described as excellent for a second number theory class. $\endgroup$ – Will Jagy Feb 25 '19 at 17:55
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Let $\theta=\frac{1+\sqrt{5}}{2}$. Then $\mathbb Z[\sqrt{5}]=\mathbb Z 1 + \mathbb Z 2\theta$ and $\mathcal O = \mathbb Z 1 + \mathbb Z \theta$. Therefore, $\left(\mathcal O:\mathbb Z [\sqrt{5}]\right)= 2$.

Equivalently, write $$ \pmatrix{1 \\ \sqrt 5} = \pmatrix{\hphantom{-}1 & 0 \\ -1 & 2} \pmatrix{1 \\ \frac{1+\sqrt{5}}{2}} $$ and note that the determinant of the matrix is $2$.

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For a quadratic number field $K=\mathbf Q(\sqrt d)$, where $d\in \mathbf Z$ is square free, you know that its ring of integers $O_K$ is the ring $\mathbf Z[(1+\sqrt d)/2]$ if $d \equiv 1$ mod $4$, $\mathbf Z[\sqrt d]$ otherwise. This can be uniformly written as $O_K=\mathbf Z[(\delta +\sqrt \delta)/2]$, where $\delta$ is the discriminant of $K$. Considering only the additive structure, $O_K$ can be viewed as a $\mathbf Z$ - module, necessarily without torsion (because included in a field), hence $\mathbf Z$ - free with $\mathbf Z$-basis {${1,(\delta +\sqrt \delta)/2}$}. Given a subring $R$ of $O_K$ which is a submodule of $\mathbf Z$-rank $2$, you ask for the index $f=(O_K : R)$. Example : $K=\mathbf Q(\sqrt 5), R=\mathbf Z[\sqrt 5]$.

Let us show that {$1, f\delta$} is a $\mathbf Z$-basis of $R$. By definition of the index, $fO_K \subset R$, so $\mathbf Z +fO_K \subset R$. But $\mathbf Z +fO_K$ has $\mathbf Z$-basis {$1, f\delta$}, hence obviously $(O_K : \mathbf Z +fO_K)=f$, and then $R=\mathbf Z +fO_K$, done. We can apply this in the opposite sense, starting from a basis of $R$ to catch the index. In your example, we get $f=2$.

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