I want to compute the class number of $K=\mathbb{Q}(\sqrt{6})$, the result should be $1$.  The discriminant $d_K$ is $24$, so the Minkowski bound is $M_K = \sqrt{6} &lt; 3$. So every ideal class contains an ideal in $\mathcal{O}_K$ with ideal norm less than or equal to $M_K$.  I know that $2$ is a norm of the principal ideal $(2 + \sqrt{6})$, in fact $2 = -N(2 + \sqrt{6})$. This is a principal ideal, so the class number is $1$.  Now I wonder if one can, after computing $M_K$, see that the class number is $1$ by knowing that $2$ is ramified (because $6 \equiv 2 \bmod 4$)? That means $2 \mathcal{O}_K = P^2$ for a prime ideal $P$.
I am calculating the class group of $\mathbb Q(\sqrt 6)$. My working is as follows:  The Minkowski bound is $\lambda(6)=\sqrt 6&lt;3$ so we only need to look at prime ideals of norm $2$. $2$ divides the discriminant, $24$, of $\mathbb Q(\sqrt 6)$, so $2$ ramifies, and so for any prime ideal $\mathfrak p$ of norm $2$, $\mathfrak p^2$ is principal. But $(2, \sqrt 6)^2=(2)$, so $\mathfrak p=(2, \sqrt 6)$ has norm $2$. If I can show $\mathfrak p$ is not principal, then I can conclude that $Cl(\mathbb Q(\sqrt 6))\cong C_2.$ But if there was an element of $\mathcal O_{\mathbb Q(\sqrt 6)}$ of norm 2, this would correspond to a solution of the diophantine equation $x^2-6y^2=2$. Clearly $x$ is even, so writing $x=2n$ we get $2n^2-3y^2=1$. Now reducing modulo $3$, we get $2n^2\equiv 1 \pmod 3$ which has no soultions. Hence $\mathfrak p$ is not principal and $Cl (\mathbb Q(\sqrt 6)) \cong C_2.$  However,  says that the class group is trivial. Where have I gone wrong? What am I misunderstanding? Is $(2, \sqrt 6)$ principal?
My wife has a Nikon D3100 that I've been using to film, but the battery typically won't last through long shoots.  I've gathered from searching around that I can't charge the camera with USB, but I wondered if there's ANY way to do that. It seems at least possible that an adapter could exist that would fill in for the battery and plug into the wall, but I haven't been able to find such a thing.  Is there such a product or technique? It would be nice to not buy a second battery, I'm really hoping for a "set it and forget it" solution. I'm using Microsoft Word 2010 and I worked in a word-2003 compatible document (.doc). After hours of working, I selected a half of text and did Ctrl+X to copy it into another document. Surprising, after closing the current document and opening a new one and pressing Ctrl+V, nothing happened. After closing original document, ms word asked me if I want to save the clipboard buffer and I pressed No, as I didn't knew that MS Word has it's own clipboard.  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Genetic linkage greater than 50 centimorgans Classically, the linkage between two loci can be measured in centimorgans (cM), which represents the percent chance that these two loci will recombine an odd number of times (generating a recombinant genotype).  Due to independent assortment, markers on different chromosomes are expected to recombine 50% of the time. As I understand, if one conducts a classic experiment and counts various progeny, then deduces the apparent linkage, values of 50 cM or more are interpreted as "impossible to determine whether on separate chromosomes or just recombines very often". See (), with grammar corrected:     The final point that we need to make regards the maximum distance that we can measure. Because of the way in which the calculations are performed, we can never have more (than) 50% recombinant gametes. Therefore the (maximum) distance that two genes can be apart and still measure that distance is just less (than) 50 cM. 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