The relation $a^n|b^n$ implies that $a|b$, where $a$ and $b$ are positive integers and $n\ge1$.
Show that $a^n \mid b^n$ implies $a \mid b$
What should be created first in a video game? I'm reading Galois Theory by Steven H. Weintraub (second edition), and finding that I'm at least somewhat short on the prerequisites. However the following proof looks wrong to me - am I misunderstanding something, or is it actually an incorrect proof?  Lemma 2.2.3. Let $F$ be a field and $R$ an integral domain that is a finite-dimensional $F$-vector space. Then $R$ is a field. Proof. We need to show that any nonzero $r \in R$ has an inverse. Consider $\{1, r, r^2, \cdots\}$. This is an infinite set of elements of $R$, and by hypothesis $R$ is finite dimensional as an $F$-vector space, so this set is linearly dependent. Hence $\sum_{i=0}^n{c_i r^i} = 0$ for some $n$ and some $c_i \in F$ not all zero.  It then goes on to show, given the above, that we can derive an inverse for $r$. However, if I consider examples like $r = 2 \in Q[\sqrt{2}]$, $r = \sqrt{2} \in Q[\sqrt{2}]$ or $r = 2 \in Q[X]/{&lt;X^2&gt;}$, the set $\{1, r, r^2, ...\}$ doesn't look linearly dependent to me. I do believe the lemma is true (and might even be able to prove it), but this does not look like a correct proof to me. Am I missing something? [Edit] Well yes, I am. Somehow I had managed to discount the possibility of any $c_i$ being negative, despite repeatedly looking at each fragment of the quoted text in an attempt to find what I might be misunderstanding. Is it possible to find all classes or interfaces in a given package? (Quickly looking at e.g. , it would seem like no.) Are battery packs allowed in hand luggage? I'd like to typeset documents using the '' of page layout. Good examples are on pages 40-53 of . The  package is helpful for this, but two additional features would be useful:   Automatic flow of text across (arbitrarily placed) columns / textpos blocks. Vertical line spacing enforced to be an integer multiple of a base line spacing, even when interrupted (for example) by an equation of arbitrary height.   Any ideas as to how either or both of these could be accomplished? I have installed Linux Bash on Windows 10 (available from Microsoft), but I've forgotten my password.  I've searched for a solution but I only find suggestions that involve doing something before the system boots, like accessing recovery mode, but that will not work in this case as this Ubuntu is just an application within a Windows system and doesn't have its own boot process.  How can I reset my password? Why did Terence Tao write Proposition 2.1.11 about mathematical induction in "Analysis I"? 9 pirates have to divide 1000 coins... A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins.  Arriving on a deserted island, they now have to split up the loot. You, as the captain of the band, have to propose a distribution plan (who gets what). What's your proposal?  Consider that this bunch is a democratic lot. If your proposal is accepted by half of the group, then everybody adheres to it. However, if folks feel you are getting greedy, and less than half of the band agrees to your proposal, then they kill you, and then your First Mate gets to make a proposal. And so it goes in decreasing order of hierarchy/seniority. What is the rigor behind U subsitution? What is the direction of stack growth in most modern systems? I am preparing some training materials in C and I want my examples to fit the typical stack model.  What direction does a C stack grow in Linux, Windows, Mac OSX (PPC and x86), Solaris, and most recent Unixes? Below, the  of second-order logic is used.    My question is about a second-order analogue of $ZFC$ other than the usual "second-order $ZFC$." Rather than define the latter, I'll just state that its (set-sized) models are exactly the $V_\kappa$s with $\kappa$ strongly inaccessible (or if you prefer, the uncountable Grothendieck universes).  Specifically, I'm interested in the second-order theory consisting of the usual (first-order) axioms of Pairing, Extensionality, Union, Choice, Infinity, and Powerset, and the Separation and Replacement schemes ranging over second-order formulas. I'll call this "$ZFC_2^{scheme}$."    Specifically, for each formula $\varphi(x, y_1,...,y_k)$ of second-order logic with only the displayed variables, each of which is first-order, we have a corresponding Separation and Replacement instance. We do not allow second-order parameters here, so formulas with second-order free variables don't yield Separation/Replacement instances.   The passage from $ZFC$ to $ZFC_2^{scheme}$ has a nice uniformity: it's an example of a more general operation $\mathcal{ZFC}$ which spits out a $ZFC$-analogue given a logic at least as strong as first-order logic.    This is the reason for the parameter restriction above: I want $ZFC_2^{scheme}$ to look just like $\mathcal{ZFC}(\mathcal{L})$ for any other logic $\mathcal{L}$, and not take into account the particular feature that we now have a new kind of variable at our disposal - and in particular, take as "syntax-free" an approach to logics as we can get away with. (This is a choice of arguable value, but it's the one I'm making for now.)   It's easy to show that every model of $ZFC_2^{scheme}$ is well-founded, so to understand its models we only need to look at transitive sets. A natural hope at this point is that $ZFC_2^{scheme}$ is just second-order $ZFC$ in disguise, that is, that for a transitive set $M$ we have $M\models ZFC_2^{scheme}$ iff $M=V_\kappa$ for some strongly inaccessible $\kappa$. However, this isn't at all obvious to me (although it is easy to show the right-to-left direction).     Question. What are the (set-sized) models of $ZFC_2^{scheme}$?   (Note by contrast that the "arithmetic analogues" $PA_2^{scheme}$ and second-order $PA$ are equivalent since each pins down $\mathbb{N}$ up to isomorphism - although that leads to .)  (I'm happy to drop Choice if that would help.)  EDIT: As Hanul Jeon pointed out below,  shows that consistently $ZFC_2^{scheme}$ (there called "$ZFC_2^{def}$") has countable models, so we have a partial answer. However, I don't see at the moment a way to get an outright $ZFC$ proof that $ZFC_2^{scheme}$ is strictly weaker than second-order $ZFC$. Finding $\frac {d(\tan \theta)}{d\theta}$ How to use PostgreSQL/PostGIS with ArcGIS for Desktop? "Users" folder shared with everyone Today I found something strange on my newly installed Windows 7 OS - my "users" folder C:\Users was shared and "Everyone" had full access... I disabled that and now it seems fine, however I'm a bit scared actually, I'd like to know what caused that and if my system is compromised. I need to make Windows 7 STOP disabling the wifi adapter when the wired LAN is connected This is a Dell Latitude E6430. There is a setting in the bios to control this behavior and I have DISABLED it but it still happens. I am also running the latest bios.  When I plug in a network cable the wifi adapter becomes disabled. I need it to stay enabled. What else could be making this happen?  EDIT: The reason I need to do this is because with NETSH WLAN commands you can create a wifi hotspot by sharing the internet connection on the wired adapter and broadcasting a ssid on the virtual wifi nic. I do this all the time but it's not working on this laptop. I'm wondering if it's a network security thing where I am. I'm going to try from my hotel tonight and report back. How to "freeze" a cloth animation at X frame? How would one freeze a simulated body at frame X and use it as a static mesh?  Following a , I've been able to do a cloth animation to my model, now I'd like to have that made static, without the use of the timeline.   The thing is, I'm modelling something that's not going to be an animation, but a static object (something like a statue/figurine). So I don't need the animation, only the shape at a particular frame that I find convenient. I'd like eventually once this is done, to also modify that deformed grid.   I've tried to follow what was said , but couldn't figure out what chipmasque was saying. I'm quite new to blender (been using it on and off, just enough to forget everything).  I'm using blender 2.63, if it is of any relevance. Didn't understand relationship 'AccountContactRelation' in FROM part of query call I'm getting the following error message:     System.QueryException: Didn't understand relationship 'AccountContactRelation' in FROM part of query call. If you are attempting to use a custom relationship, be sure to append the '__r' after the custom relationship name. Please reference your WSDL or the describe call for the appropriate names.   Select Email, Id, (Select ContactId, AccountId From AccountContactRelation) From Contact  In AccountContactRelation object there are two lookup fields for Account and for Contact.  Please advise how to avoid such error message? How do Prestige Classes that advance spellcasting interact with racial spellcasting ability? For example Lets use the Marrutact from Sandstorm, it can cast as a 5th level wizard. if it then takes a prestige class that requires the ability to cast 1st level Arcane spells and advances arcane casting such as Abjurant Champion. assuming it meets all other prerequisites does the Marrutacts spellcasting improve when it takes a level in Abjurant Champion or does it gain no benefit? At the moment I'm working with the . For this I read the following paper: .  But I have some mathematical problems with this formalism. I don't really see why this is a variation principle since it is opartor-valued.   The action principle is defined by: $\delta S  = G_1 -G_2$, where $S$ is the action operator $S = \int_{t_1}^{t_2} dt ~L$ and $G$ the generator operators of the transformations.  What does it mean that the action principle is operator-valued? For the classical action where $S$ is real we search for a stationary point but for operators? I also have a problem that $\delta S \neq 0$. What interpretation has this additional term $G_1 - G_2$? Can't update because the word prox* is blocked