Find the ring of algebraic integers in $K=\mathbb Q(\sqrt[3]{2})$.   So, I know that $K=\{a+b\sqrt[3]{2}+c\sqrt[3]{2}^2 \mid a,b,c \in \mathbb Q\}$.   My professor has done very little on this topic. I know that an algebraic integer is the root of a monic polynomial with integer coefficients.   I have no clue on where to start with this question. Any help is much appreciated!
This seems to be one of those tricky examples. I only know one proof which is quite complicated and follows by localizing $\mathbb{Z}[\sqrt[3]{2}]$ at different primes and then showing it's a DVR. Does anyone know any simple quick proof?
Keep iphone active while running program how to set iPhone device to stay active ( to not lock ) while my app is running ? Any idea How to apply for a Schengen visa when the travel dates are uncertain My husband has a Chinese passport and I am a U.S. citizen; I don't need a visa to travel to Schengen countries. I am helping him to apply for a visa.  Our plans are to go to Paris, Italy and Spain, but we haven't nailed down definite dates; the approximate dates are August 18 to September 11. We made an appointment with Spain consulate to apply for the visa. We need to show air flight reservations, I could book it online without any problems.    Since we don't have definite dates, what dates should he put down on the application?  I know that, once they approve you, they give you 90 days, so does it matter what dates I put down on the visa application?   Our appointment at the Spanish Consulate is in a few days. Do we know of any living forms on Earth that could continue to live on Mars? Prove $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges if $\sum_{n=1}^{\infty}a_{n}^{2}$ and $\sum_{n=1}^{\infty}b_{n}^{2}$ converge This is a homework problem for an undergrad topology course.  Let $l^{2}$ be the set of all real-valued sequences $(c_{n})$ where $\sum_{n=1}^{\infty}c_{n}^{2}$ converges. Let $(a_{n}),(b_{n})\in l^{2}$.  Claim: $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges.  I've done a couple of examples where $a_{n}$ and $b_{n}$ are general harmonic series. I've also written out the definition of convergence for $a_{n}$ and $b_{n} $. My problem is that thus far I do not understand on any level why my claim is true. I'm looking for a shapefile containing an 1000m or 2000m UTM grid ("graticule"). All I was about to find on the web is a 10000m UTM grid (here: ) and I am aware that a 1000m UTM grid for the whole world might be quite big.  That's why I'm now wondering if and how it would be possible to generate such a shapefile for a part of the world. I would prefer to use open source tools like gdal. Others output formats than a shapefile might be ok too.   Has anyone an idea? What is there to do after beating the Elite Four/Champion (Post-Game)? I've defeated the Elite Four and the Champion. The credits rolled and I've been teleported to my home in Vaniville Town. What is there to do now? Any specific events/features that are unlocked now?  I already know of and caught Mewtwo and Zyrgarde. Is this proof of $\liminf E_k \subset \limsup E_k $ correct? I am wondering if my proof is correct? Thank you for whoever willing to take a look at it for me.     Proof $\liminf E_k \subset \limsup E_k $   If $\{E_k\}_{k=1}^\infty$ is a sequence of sets, we define \begin{align*} \limsup E_k &amp; = \bigcap_{j=1}^\infty\left(\bigcup_{k=j}^\infty E_k\right)\\ &amp;= \bigcap_{j=1}^\infty(E_j \cup E_{j+1} \cup \cdots)\\ &amp;= (E_1 \cup E_2 \cup \cdots) \cap (E_2 \cup E_3 \cup \cdots) \cap \cdots. \end{align*}  Therefore, $\limsup E_k$ consists of those points in $\mathbb{R}^n$ which belong to infinitely many $E_k$. \begin{align*} \liminf E_k &amp; = \bigcup_{j=1}^\infty\left(\bigcap_{k=j}^\infty E_k\right)\\ &amp;= \bigcup_{j=1}^\infty(E_j \cap E_{j+1} \cap \cdots)\\ &amp;= (E_1 \cap E_2 \cap \cdots) \cup (E_2 \cap E_3 \cap \cdots) \cup \cdots. \end{align*} Therefore, $\liminf E_k$ consists of those points in $\mathbb{R}^n$ which belong to all $E_k$ for $k \geq k_0$.  Now consider $x \in \liminf E_k$, then $x \in E_1 \cap E_2 \cap \cdots$, or $x \in E_2 \cap E_3 \cap \cdots$ and so on, that is to say, $x \in E_{k_0} \cup E_{k_0+1} \cup \dots$ for some $k_0$. More specifically, $x \in E_k \forall k \geq k_0$. Therefore, $x \in (E_1 \cup E_2 \cup \cdots) \cap (E_2 \cup E_3 \cup \cdots) \cap \cdots$, so we showed $x \in \limsup E_k$. Describe the units in $\mathbb{Z}[\sqrt{-d}]$ Why is scale factor inversely proprotional to temperature? In all lectures, books and papers I have read about, the scale factor of the universe is inversely proportional to temperature  $$a \propto \frac{1}{T}$$  What is the reasoning behind this relation?  People did mention stretching of wavelength proportional to scale factor and wavelength being inversely proportional to temperature by Wein's law, but somehow this argument doesn't seem convincing to me. How did we initially come about to this relation? I am wondering if the explanations correct and whether what is the difference between different &quot;seems&quot;.   He seems happy. He seems a happy fellow. He seems to be a happy fellow. He seems to work hard. He seemed to be happy   &quot;Seem&quot; also enters into an 'impersonal' construction which employs 1) the 'dummy' subject it + an active tensed form with &quot;seem&quot;, followed by 2) a that clause (sometimes called a 'content' clause) which expresses the assertion which is hedged or qualified.   It seems that he is happy (but he may not be).  It seemed that he was a happy fellow (but he wasn't).  It will seem that he is working hard (but he will in fact be spending his time answering questions on ELL instead of working).  He is seemed to be something. How can I create and extract zip archives from the command line? Counting degrees of freedom of gauge bosons Gauge bosons are represented by $A_{\mu}$, where $\mu = 0,1,2,3$. So in general there are 4 degrees of freedom. But in reality, a photon (gauge boson) has two degrees of freedom (two polarization states).   So, when someone asks about on-shell and off-shell degrees of freedom, I thought they are 2 and 4. But I read that the off-shell d.of. are 3.   And my question is how to see this? I have my A* pathfinding algorithm right now working such that given a graph it can find the route from one starting node to one goal node.   This works well for many problems, but sometimes I find that multiple goal nodes are valid.  For example, in a game, an NPC might have to go to a switch to shut a door. If there are multiple switches that shut the door in question, it doesn't matter which switch it goes to - it just needs to go to one of them.  How can I augment my pathfinding algorithm to find the path to the best goal (the shortest path)? As in, the algorithm searches for multiple goals and when it finds just one, it returns the path to that one.  One solution would be to run the algorithm as many times as there are valid goals, but you can guess why that might be inefficient. Running a .desktop file in the terminal From what I can gather, .desktop files are shortcuts that allow application's settings to be customized. For instance, I have lots of them in my /usr/share/applications/ folder.   If I open that folder in nautilus, I can run these applications just by double clicking its associated file, e.g. double-clicking firefox.desktop runs Firefox. However, I can't find a way to do the same thing via terminal.   If I do gnome-open foo.desktop it simply opens foo.desktop as a text file. If I make it executable and then run it in bash it simply fails (which is expected, it's clearly not bash script). EDIT: Doing exec /fullpath/foo.desktop gives me a Permission denied message, even if I change ownership to myself. If I make executable and do the same command, the terminal tab I'm using simply closes (I'm guessing it crashes). Finally, if I do sudo exec /fullpath/foo.desktop, I get an error reporting sudo: exec: command not found.  That's my question, how can I run a foo.desktop file from the terminal? Loading a .X model in DirectX 9 Add a transparent image on top of a material I initially thought that dark energy must in some way violate conservation of mass and energy since the component of the energy density of space that comes from dark energy is constant, and space is expanding. Therefore, as space expands the energy in the universe that comes from dark energy would increase. I presumed the source of this energy was not coming from the conversion of other types of energy to dark energy, so it must violate conservation.  I decided to Google this and came upon this article:    It says that dark energy does NOT violate conservation and quotes Carroll, Press, and Turner (1992):  "…the patch does negative work on its surroundings, because it has negative pressure. Assuming the patch expands adiabatically [i.e. without loss or gain of heat], one may equate this negative work to the increase of mass/energy of the patch. One thereby recovers the correct equation of state for dark energy: P = – ρ c2. So the mathematics is consistent."  Is there a way to explain this in layman's terms? (The blog attempted to do this, but it was very unclear to me.) More specifically, can you explain where my initial train of thought described above fails when I erroneously concluded that dark energy violates conservation?  Thank you. What could cause a toilet to leave the flapper up and keep running? I have an American Standard toilet I installed last year. I don't know much about plumbing, but I was able to install this toilet. It seems to function okay, except for what I think is a mechanical problem.  Sometimes when the toilet is flushed, the flapper doesn't close and water keeps running.  The handle feels like it's slightly stuck in the up position.  I looked inside the tank once when it was doing this, and the chain attached to the handle was slack. I don't understand why the flapper isn't closing.  I've tried putting more slack in the chain and loosening the plastic nut that holds the handle on the toilet, but it doesn't seem to be helping.    How can I fix this? Why does $(\frac{p-1}{2}!)^2 = (-1)^{\frac{p+1}{2}}$ mod $p$? Assume $p$ is prime and $p\ge 3$.  Through experimentation, I can see that it's probably true. Using Wilson's theorem and Fermat's little theorem, it's equivalent to saying $2^2 4^2 6^2 \cdots (p-1)^2 = (-1)^{\frac{p+1}{2}}$ mod $p$, but I can't figure out any more than that. How did Bill and Percy get 12 OWLs? In Prisoner of Azkaban, Hermione took 12 subjects, and needed a Time-Turner to do so. However, it is mentioned in the series that both Bill and Percy Weasley got 12 OWLs, so would have taken 12 subjects as well, so both would have needed Time-Turners for three years to do this. Doesn't this seem a little unusual, given how strict the Ministry is over the use of Time-Turners,  that they would give them to several Hogwarts students?