How could I prove that $\Bbb Q({\sqrt2},{\sqrt3})$ and $\Bbb Q(2^\frac{1}{4}))$ aren't isomorphic? I need to prove it and I don't know how, but it seems easy. Thanks everybody.
I wish to show that $\mathbb{Q}(\sqrt2, \sqrt3)$ is not isomorphic to $\mathbb{Q}(\sqrt[4]2)$.    What I did so far was to show that $\mathbb{Q}(\sqrt2,\sqrt3) =\text{span}\{1,\sqrt2,\sqrt3,\sqrt6\}$.  Assuming they are isomorphic, denoting $f$ the isomorphism, then $f(1) = 1$, so $x^2-3$ has a root in $\mathbb{Q}(\sqrt2,\sqrt3)$ so $\sqrt 3\in\mathbb{Q}(\sqrt[4]2)$. ($\sqrt2 \in \mathbb{Q}(\sqrt[4]2)$, $\sqrt[4]2\cdot\sqrt[4]2=\sqrt2)$.   Thus under the assumption, $\mathbb{Q}(\sqrt2, \sqrt3)\subseteq \mathbb{Q}(\sqrt[4]2)$. As a vector spaces of the same dimension over $\mathbb{Q}$, it means that $\mathbb{Q}(\sqrt2, \sqrt3)= \mathbb{Q}(\sqrt[4]2)$.  Now all is left to do it to show a contradiction: $\sqrt[4]2\in\mathbb{Q}(\sqrt2,\sqrt3)$, then $\sqrt[4]2 = a+b\sqrt2+c\sqrt3+d\sqrt6$ so $2=(a+b\sqrt2+c\sqrt3+d\sqrt6)^4$, I believe that opening the expression, I would find out that it can't be an element of $\mathbb{Z}$. However this is a really inelegant way (if it really works), and I wish for a better way to do so.  An other approach: $\mathbb{Q}(\sqrt2,\sqrt3) = \mathbb{Q}(\sqrt2 + \sqrt3)$ by the irreducibility of $x^4-10x^2+1$ over $\mathbb{Q}$ (an some dimension theorem usage). Also, $x^4-10x^2+1$ splits in $\mathbb{Q(\sqrt2, \sqrt3)}$.   Is there a way to show that this polynomial doesn't split over $\mathbb{Q}(\sqrt[2]4)$? with different method then assuming that $\sqrt3\in\mathbb{Q}(\sqrt[4]2)$ and getting a contradiction? Maybe the fact that $\sqrt[4]2$ has complex roots might help?
Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous map such that $\overline{f(\mathbb{N})} = \mathbb{R}$. Prove that $f$ is surjective.  I tried to use the definition of the continuity and $\overline{f(\mathbb{N})} = \mathbb{R}$ to show that $f$ is surjective but it didn't work. Since $f$ is continuous, the preimage of every open set is open. Also it results in $f(\overline{A})\subseteq \overline{f(A)}$ for every set $A$. How this couples with $\overline{f(\mathbb{N})} = \mathbb{R}$ to show surjectivity? TikZ \foreach loop with macro-defined list I have a small problem with the TikZ \foreach loop whenever the list, over which we loop, is defined my a macro. The following code example will show my problem:  \documentclass{scrartcl} \usepackage[utf8]{inputenc} \usepackage{tikz}  \begin{document} Picture one:  \begin{tikzpicture} \foreach \x/\y in {1.0/2.0, 3.0/4.0} \node[draw] at (\x,\y) {\x--\y}; \end{tikzpicture}  Picture two:  \newcommand{\mymacro}{1.0/2.0, 3.0/4.0} \begin{tikzpicture} \foreach \x/\y in {\mymacro} \node[draw] at (\x,\y) {\x--\y}; \end{tikzpicture} \end{document}   The expected result is that both images should look the same. But in the second image, I only get one node - and both \x and \y are each time expanded to 1.0/2.0, 3.0/4.0. I hope someone can help. Immediately, his insides started writhing as though he'd just swallowed live snakes -- doubled up, he wondered whether he was going to be sick -- then a burning sensation spread rapidly form his stomach to the very ends of his fingers and toes -- next, bringing him gasping to all fours, came to a horrible melting feeling, as the skin all over his body bubbled like hot wax -- and before their eyes, his hands began to grow, the fingers thickened, the knuckles were bulging like bolts -- his shoulders stretched painfully and a prickling on his forehead told him that his hair was creeping down toward his eyebrows -- his robes ripped as his chest expanded like a barrel bursting its hoops -- his feet were agony in shoes four sizes too small      Harry potter and the chamber of secrets   It is the description of the feeling of drinking Polyjuice Potion, which makes someone into a different person, from the point of view of Harry Potter turning into Gregory Goyle.  And I don't understand what 'bring him gasping to all fours' means, especially 'four'. Does four here mean number 4? Minimum requirements of Ubuntu 20.10 Does anyone know what are the minimum requirements to install Ubuntu 20.10? Hahn-Banach Theorem for separable spaces without Zorn's Lemma Basis of the null space Find the basis of the null space for the matrix  $$D = \begin{bmatrix}0 &amp; 0 &amp; 0 &amp; 0\\0 &amp; 1 &amp; 1 &amp; 0\\1 &amp; 1 &amp; 0 &amp; 0\\1 &amp;1 &amp; 0 &amp; 0\end{bmatrix}.$$  I computed the solution by letting eigenvectors $(x,y,z,w)$ such that $y+z=0,x+y=0$, and I got a basis vector $\{(1,-1,1,0), (0,0,0,1)\}$ . But from one textbook  I have found the solution as $\{(1,0,0,0),(0,0,1,1)\}$. May I made a mistake? Thanks! Some Data Failed to Reconstruct, Motion Tracking I added tracks and stuff, but then when I tried to solve motion track, there was a error that said:     Some Data Failed to Reconstruct   and it also said to look into the console for details, which I did. This is what the console said:    By my understanding, my video is 372 frames long, and Blender is cutting the footage at 318 for some reason. Here is the image of the scene:    Why? Here are links to  and also . Also the next morning I did a camera solve, but then most of the frames went red. I have seen a "Universal Laptop Adapter" sold in a computer shop before. I am thinking of getting one for a laptop. Is it safe to use? What differences are there between one of these and a typical laptop adapter? Fastest method to determine my PC's IP address (Windows) This may seem trivial but I'm looking for the quickest method to determine my PC's IP address within my network.   This changes regularly as I connect from one network to the next via DHCP The connection changes from wired to wireless Windows 7 PC (although ideally the perfect solution would work on all versions of Windows) Various adapters installed (e.g. VMWare, Tunnels, etc.)   I need to know this as I often run a local web server where I need to access files over the local network... and since it changes regularly I want the fastest method to retrieve the address.  My current flow is:   WindowsKey + R to open the Run dialog "cmd" + Enter to open a command prompt "ipconfig" + Enter to get the diagnostics info Scroll to or resize the window so that I can see the line in my Ethernet adapter Local Area Connection for my ipv4 Address Remember it to type elsewhere... or  Right click > select Mark > highlight the address > press Enter to copy it to the clipboard   This wasn't so bad with Windows XP when I had no additional adapters, tunnels, wireless connections etc. but the amount of data returned with this command makes it hard to pluck out.  Surely there must be a better, faster way! (bonus points if adding it to the clipboard is easily accomplished) There a  who has no valid parent user. It is indicated that his chat account is associated to a Maths.SE account, which does not appear to exist. The user was still able to chat. Is it normal? If it's a bug, could someone look into it?  Edit: if you browse the Google cache for users named Skullpatrol, it turns out a single series of associated accounts. The user is now named . I don't know what went wrong, maybe an account merge or something like that, but it needs someone from staff looking into it with some superpowers. I also tried to update the parent user of the chat account to this &quot;Rob&quot; profile, but the interface goes all buggy on me (works for other users, fails only for this one): To avoid printing single letters at varying sizes and measuring how far away I can see it from a distance is there a formula for determining what size font for a particular read distance? Is it better to take the midpoint between the ascender line and x-height or to go purely off of the x-height (since going purely off the ascender line could leave lowercase letters difficult to read)? How does vertical height play into this? Would I need to then figure the hypotenuse distance instead of a straight distance?  For example if I'm shooting for a 20' read distance but the sign is to be placed about 30' up should I use a distance of  35' instead of 20'?  I'm shooting for a 20' read distance for the headline and then possibly a 10' read distance for additional content depending on how it all fits on the active area. Building a matrix by merging the same row vector multiple times In the annual rep league, why isn't the difference between rep earned this year and total rep always the same? In the  here on Meta Stack Exchange, it shows that I've earned 9,876 rep this year, and 10,359 rep overall. (It's updated every day at 0:00 UTC, and I so far have earned 1 rep today, since an answer I downvoted a while ago was deleted.)  The difference between my total rep and the rep I earned this year is 483, which means 483 of the rep I've earned is rep I earned in 2017 or earlier. This amount normally never changes, though I can see some reasons why it might change, such as if rep is recalculated and something affects a rep event occurring in a previous year (e.g. a vote from last year was invalidated due to user deletion, a post I made at that time was deleted or undeleted, etc.)  This amount, 483, was a recent change; it used to be 461. Ordinarily, I'd chalk this up to a rep event from last year being recalculated, but I've gone through my full rep history and I can see nothing of the sort happening. On the other hand, on the same day that this number changed from 461 to 483,  was undeleted, recalculating my rep and giving me back 22 (which happens to equal 483-461). It therefore makes sense if that post was from a previous year. However, the post is from this year, so it doesn't make very much sense.  Is there a bug in the way yearly rep in the rep league is calculated, in that rep regained from an undeleted post made the same year is not re-added to the rep earned this year (while it is subtracted properly upon deletion)? Or is there some other event that occurred in the three weeks before 2018 that I had my account that caused this difference to change, that I'm not seeing, and the change being equal to that is just a coincidence? Using Unicode Zero Width Spaces allows all space comments Is there a rule of thumb for objective questions asked out of curiosity? How do I disable negative scaling? By default blender allows you to scale past 0 - how do I prevent this?  i.e. it's common in other 3D modelling software to resize a set of vertices against an axis to align them - however this isn't possible when negative scaling is enabled. Construct ideals in $\mathbb Z[x]$ with a given least number of generators This has been , but a long time ago, and got closed for a rather confusing reason (converting comments to answers is not an "exact duplicate" of converting answers to comments), so I'm going to bring it up again.  I think it makes sense to have a user (not asker) option to convert a comment they posted on a question to an answer on the question, if it turns out it led to the asker figuring out the answer. We've all seen askers reply to comments with "@foo Hey, that was it! Post that as an answer and I'll accept it"; why not make it simple for the user to do that? It seems trivial to implement, and there's no formatting issues like there is going from answers to questions. I'm not sure how it could be "abused", because you need less rep to answer a question than to comment on it. I don't know if the answer's post time should be the time of the comment or the time it was converted; I think probably the latter. I also don't know if the comment should be deleted, left alone, or replaced with a link to the answer; again I think probably the last Pattern in Isomorphisms between $\mathbb Z_m \times \mathbb Z_n$ and $\mathbb Z_{mn}$ In general, the group $\mathbb Z_{mn}$ under addition $\bmod mn$ is isomorphic to the direct product $\mathbb Z_m \times \mathbb Z_n$ if and only if $\gcd(m,n)=1.$ This is easy to see since $\mathbb Z_{mn}$ is generated by the element $(1,1)$, and if $\gcd(m,n)=1$, then this will only be zero if it is multiplied by itself $mn$ times (resulting in the cyclic group).   However I'm interested in constructing an isomorphism between the two groups as an explicit expression. I've been playing around a bit with some examples, one of which is the function $\phi : \mathbb Z_2 \times \mathbb Z_3 \to \mathbb Z_6$ where $\phi(a,b) = 3a + 4b \pmod 6$, which gives us the desired isomorphism. I'm interested here in the relationship between the integers $2$, $3$, and the coefficients $3, 4$ of $a$ and $b$. After playing around in Wolfram|Mathematica, I managed to obtain the following isomorphisms $\phi_{m,n} : \mathbb Z_m \times \mathbb Z_n \to \mathbb Z_{mn}$ for coprime $m,n$ in $\{1, 2, \dots, 10\}$.   First of all, the following results seem to hold in general.  $$\phi_{1, n}(a, b) = a + b \qquad \text{for all $n$}$$  $$\phi_{m, 1}(a, b) = a + b \qquad \text{for all $m$}$$  $$\phi_{2, n}(a, b) = na + (n + 1)b \qquad \text{for all $n$}$$ $$\phi_{m, n}(a, b) = K a + L b \iff \phi_{n,m}(a, b) = L a + K b \qquad \text{for all $m, n$}$$  Beyond this point, I could not spot a general pattern. The isomorphisms I obtained are:  $$\phi_{3, 4}(a, b) = 4a + 9b$$ $$\phi_{3, 5}(a, b) = 10a + 6b$$ $$\phi_{3, 7}(a, b) = 7a + 15b$$ $$\phi_{3, 8}(a, b) = 16a + 9b$$ $$\phi_{3, 10}(a, b) = 10a + 21b$$ $$\phi_{4, 5}(a, b) = 5a + 16b$$ $$\phi_{4, 7}(a, b) = 21a + 8b$$ $$\phi_{4, 9}(a, b) = 9a + 28b$$ $$\phi_{5, 6}(a, b) = 6a + 25b$$ $$\phi_{5, 7}(a, b) = 21a + 15b$$ $$\phi_{5, 8}(a, b) = 16a + 25b$$ $$\phi_{5, 9}(a, b) = 36a + 10b$$ $$\phi_{6, 7}(a, b) = 7a + 36b$$ $$\phi_{7, 8}(a, b) = 8a + 49b$$ $$\phi_{7, 9}(a, b) = 36a + 28b$$ $$\phi_{7, 10}(a, b) = 50a + 21b$$ $$\phi_{8, 9}(a, b) = 9a + 64b$$ $$\phi_{9, 10}(a, b) = 10a + 81b$$ If you want to try and generate a few yourself, here is the Mathematica function I implemented. I'm sure it can be optimised, but for the first 10 coprime integers it sufficed.  isomorphism[{m_, n_}] :=       Module[{x = Null},       For[i = 1, i &lt; m*n,           i++, (If[#[[2]] == Table[i, {i, 0, m*n - 1}],           x = ToString[#[[1]][[1]]] &lt;&gt; "a+" &lt;&gt; ToString[#[[1]][[2]]] &lt;&gt;          "b"; Break[]] &amp;) /@       Table[{{i, j},       Mod[i #[[1]] + j #[[2]], m*n] &amp; /@       Table[{Mod[i, m], Mod[i, n]}, {i, 0, m*n - 1}]}, {i, 1,      m*n}, {j, 1, m*n}][[i]]]; x];   I'd appreciate if anyone can help me find the pattern to determine $\phi_{m, n}$ in general. Can I find the connected components of a graph using matrix operations on the graph's adjacency matrix? If I have an adjacency matrix for a graph, can I do a series of matrix operations on the adjacency matrix to find the connected components of the graph?