Is  $ \mathbb{Q}( \sqrt[3] 2)$ is  a normal extension of $\mathbb{Q}?$ My attempt : I think yes By definition  of normal extension : $L/K$ is a normal extension if and only  if  for every irreducible  polynomial  $f \in K[x]$  with at least  one root in $L$ Now  i take $f(x)=x^3-2$ then $f(x)$ has one root i,e $x^3=2 \implies x=\sqrt[3] 2$ So here we have atleast one root in $\mathbb{Q}$
In a question related to introductory Galois theory, I was asked to given an example of a tower of fields $F \subset K \subset E$ such that $E$ is a splitting field for some polynomial $f(x) \in F[x]$, but that $K$ need not be a splitting field. The example that comes to mind for me is $\mathbb Q \subset \mathbb Q(\sqrt[3]{2}) \subset \mathbb Q(\omega_3, \sqrt[3]{2})$.  So I know that the minimal polynomial for $\mathbb Q(\omega_3, \sqrt[3]{2})$ over $\mathbb Q$ is $x^3-2$. This polynomial does not split in the intermediate field $\mathbb Q(\sqrt[3]{2})$.  I wanted to make sure that I'm justifying that this intermediate field is not a splitting field correctly. Now I may be confused about the books terminology. When we say something is not a splitting field, we always have to refer to a specific polynomial correct? I've read a term called a normal extension which I think refers to one in which every polynomial splits, but my book doesn't mention these.   If splitting field means with respect to the polynomial $x^3-2$, then I've already demonstrated that $\mathbb Q(\sqrt[3]{2})$ is not a splitting field. Also, it couldn't be a normal extension either because that polynomial doesn't split.  Does it sound like I'm understanding this correctly or are there some subtleties I might be missing?
Domain Controller etiquette after a crash between Server 2003 & Server 2008 I have 2 domain controllers on a non-Internet connected network running Windows Server 2008 R2. They running RAID 5 on a 8 Hard Drive server. Sometime last week the Boot Sector became corrupt on DC1 and I was unable to recover the server. I am not going to have to rebuild the server. I have been reading about the changes from Server 2003 to Server 2008. DC1 was the main domain controller and now that it won't but it is all falling to DC2. I know that in previous versions of Windows Server you had to make on the primary and one the secondary. Since DC2 does not see DC1 anymore is there anything I need to do to DC2 as it is now the one in control? I am thinking some kind of meta data clean up but I can't find any hard information on what procedure to do. I understand as of Server 2008 domain controllers are on an equal footing and provide replication to each other so they should be the same correct? Born's Rule, What is the Reason? What is the problem that gets worse after you try to solve it? Example sentence - This problem is a _____ which gets complicated every time you try to solve it.   Is there such a word that represents a problem which will become worse if/when one tries to solve it?  Note - After seeing the comments and answers I'd like to add that this is not a can of worms or Pandora's box because it doesn't create a new problem but just worsens.   Example: Magic leaves protecting a fruit by enclosing it. Every time a leaf is touched, all the leaves close together further, that is more tight than before. Is there a single word with the same meaning as "good moments" or "pleasant memories"? I'm looking for a word which describes good thing(s) that happened to someone. The only word I can think of is "joy".  Some examples might be:   Wayne has many pleasant memories of the days spent with the family. She could remember many good moments from the last weekend. Currently my spouse and I rent a small apartment.  Eventually we'd like to expand the family, and a one bedroom rental won't cut it forever - even if we continue renting we'll be expanding to a detached single family home eventually.  In our area the rent ratios work out to about 16, which is under the cut for buying being a better financial decision over the long haul - i. e. I don't think it'd be economically an awful decision to buy (NYT calculator shows 10 years at the current prices and rents for buying to make sense).  I've also read enough of the rental propagandists to know that home ownership isn't the best way to grow my money.  My reasons for wanting to own are the intangibles: knowing we aren't going to have to move due to a landlord, being able to modify the place to our own liking, and that intangible sense of "this is mine" (even when it's the bank's for a while).  The things we do have:   Around (at or over) a 30% down payment depending on purchase price, and we won't be under 30% Closing costs separate from that fund Around 1% of the home value in reserve for emergency repairs separate from those funds A six month emergency fund separate A fund to replace an ailing car that will probably die before we can save up after this expense   At some point in the year or so after the purchase we plan to have enough saved up to pay down the principal and change our monthly payment somewhat, but not significantly.  So it seems like we've got the financial ducks in a row.  We can continue to fully fund our retirement accounts and make the monthly payments (PITI).  The only hesitation we have is that we can't make our payments comfortably on one income alone - it'd be over 30% of one income to do so, by just a bit.  So neither of us can completely stop working, although the second income does not need to be large.  Are we missing any important financial data about this step?  Should we be concerned about job loss (even though no signs of it are imminent for either of us)? Is it okay to use Stack Overflow as the support forum for a product or project? I recently saw a question about a problem (or possible bug) with some product on Stack Overflow. At first, I wanted to suggest that the asker post his question to that product's support forum.  When searching for the URL of that forum, I noticed that the product developer closed its support forums and now advises users to post all their questions on Stack Overflow instead.  Is this within the intended use cases of Stack Overflow, or is this abuse? I am writing an application that if the user hits back, it may resend the same information and mess up the flow and integrity of data. How do I disable it for users who are with and without javascript on? What happens when one is affected by something such as energy drain, and permanently loses levels when one only has racial HD? Does the victim of the spell die? Or do they permanently lose racial HD? Anti-Cheat for MMOs issues and solutions? Most online games suffer for this problem and there are some  out there to cover this but recently from what I've seen most companies are trying to avoid those programs creating their own system (blizzard with warden, aion with their own checking, etc...) but it is a very difficult case to prevent such ...  Just wanted to say i made another question topic as community wiki specific for posting applications and it is features and the ideas behind this question here is not to list those but instead to know more about what were your attempts, issues along the road and possible solutions, piece of codes if application and so forth.  What sort of protection do you guys use in order to (try to) prevent such acts:   Anti-cheat Software ? (please state which, why and possible problems you faced, for example aion stopped using GameGuard at beta stage due to a huge amount of compatibility problems and complains from their users) Your own code ? (what sort of verification do you usually run to make sure of the bad-faith players, for example speedhack, verify new location against speed+last location, etc...) None, i dont care ? (any personal reason or input you did like to share ?)   Any interesting piece of code related to the issue you would like to post ?  Feel free to rework my text my english inst that great  ?** Beamer - Color of header foreground with AnnArbor theme I use beamer with the theme AnnArbor and colortheme whale.  I tried to use:  \setbeamercolor{section in head/foot}{fg=white, bg=blue}   or  \setbeamercolor{palette tertiary}{fg=white, bg=blue}   But for an unknown reason, the bg works but not the fg... The sections names stay black in the header.    EDIT : the problem comes from the hyperref package.  This code works well :  \documentclass{beamer} \usetheme{AnnArbor} \usecolortheme{whale} \setbeamercolor{frametitle}{fg=white,bg=blue}  \begin{document}     \section{Test 1}     \subsection{Test 1.1}     \begin{frame}         \frametitle{Test}     \end{frame} \end{document}   Whereas this one doesn't works :  \documentclass{beamer} \usetheme{AnnArbor} \usecolortheme{whale} \setbeamercolor{frametitle}{fg=white,bg=blue} \usepackage{hyperref} \hypersetup{     colorlinks=true,     linkcolor=black,     urlcolor=blue,     citecolor=gray,     frenchlinks=false,     pdfborder={0 0 0} }  \begin{document}     \section{Test 1}     \subsection{Test 1.1}     \begin{frame}         \frametitle{Test}     \end{frame} \end{document}   And that's pretty normal because of this line :  linkcolor=black, How do I change text on the submission button in the node form? Is there a way to determine the device running an application. I want to distinguish between iPhone and iPod Touch, if possible. Saving high-quality graphics through commands Convert UTC/GMT time to local time We are developing a C# application for a web-service client. This will run on Windows XP PC's.  One of the fields returned by the web service is a DateTime field. The server returns a field in GMT format i.e. with a "Z" at the end.  However, we found that .NET seems to do some kind of implicit conversion and the time was always 12 hours out.  The following code sample resolves this to some extent in that the 12 hour difference has gone but it makes no allowance for NZ daylight saving.  CultureInfo ci = new CultureInfo("en-NZ"); string date = "Web service date".ToString("R", ci); DateTime convertedDate = DateTime.Parse(date);               As per :     UTC/GMT Offset       Standard time zone: UTC/GMT +12 hours   Daylight saving time: +1 hour   Current time zone offset: UTC/GMT +13 hours     How do we adjust for the extra hour? Can this be done programmatically or is this some kind of setting on the PC's? Let $x_0 \in \mathbb{R}$ and $f,g: \mathbb{R} \to \mathbb{R}$  such that $f$ is discontinuous at $x_0$ but $g$ is continuous at $x_0$, then $f+g$ at $x_0$ is...   My approach: After some graphical attempts to construct such a pathological case as given above I claim that the described scenario isn't possible. That is $$f \text{ disc. at } x_0 \text{ and } g \text{ cont. at } x_0 \implies f+g \text{ disc. at } x_0 $$  Proof: I was trying to come up with a contradiction, because if they happen to work out I find these kind of proofs to be most satisfying.   Assume $f+g$ is continuous at $x_0$ that is $$\forall \epsilon &gt; 0 ,\exists \delta_1 &gt; 0 : \forall x \in \mathbb{R}, |x-x_0| &lt; \delta_1 \implies |f(x)+g(x)-f(x_0)-g(x_0)| &lt; \epsilon  $$  Since this statement must be true for all $\epsilon &gt; 0$ I could choose $\epsilon:=1$. Next I know that also $g$ is continuous at said point $x_0$, thus  $$\forall \epsilon &gt; 0, \exists \delta_2 &gt; 0 : \forall x \in \mathbb{R}, |x-x_0| &lt; \delta_2 \implies |g(x)-g(x_0)| &lt; \epsilon $$  and I have that $f$ is NOT continuous at $x_0$ therefore: $$\exists \epsilon^* &gt; 0, \forall \delta&gt;0: \exists x \in  \mathbb{R}, |x-x_0| &lt; \delta \wedge |f(x)-f(x_0)| \geq \epsilon $$  My problem is to bring all these three statements nicely together. Mainly I get confused by the quantors. In order to match all three statements I assume I have to choose $\epsilon &gt; 0$ such that the last statement is met, so say $\epsilon:= \epsilon^*&gt;0$, so statement 1 and 2 are still true for said $\epsilon$  I also need to choose $\delta$. Lets say $\delta = \min{(\delta_1,\delta_2)}$ now I should be able to work with all three statements at the same time right? But then I stumble with my argumentation.  Let $|x-x_0|&lt; \delta$, then we obtain $$ |f(x)+g(x)-f(x_0)-g(x_0)| \leq |f(x)-f(x_0)| + |g(x)-g(x_0)| &lt; - \epsilon + \epsilon =0 $$ which means that $|f(x)+g(x)-f(x_0)-g(x_0)| &lt;0$ and I hope that this is a contradiction because $|.| \geq 0$. Please comment (or answer) if I am right or absolutely on the wrong path. I'd also appreciate it if you criticize my formulation. Oxford Comma Conventions According to the Wikipedia page for the , "Use of the comma is consistent with conventional practice" and "Use of the comma is inconsistent with conventional practice." Did the Oxford Comma come before its omission, or was the Oxford Comma traditionally omitted?  It makes logical sense that every item in a list would be separated in the same way: by a comma. If the Oxford Comma is conventionally correct, when and why did people begin to omit it? CPU cooler does not fit, will it cool worse if I cut tips of heatpipes off? Case: zalman z9 neo  Cooler: deepcool gammaxx 400  According to specs it should fit, but it does not, probably because my motherboard is mini-itx and cooler located a bit to the left  I'm thinking that cutting off those tips of heatpipes on the left would solve that; if I do that, will cooler work worse, like heatpipes would transfer heat not as good or something? In the 70's I had a stack of old Amateur Radio magazines (50s-60s), and for a long time I saved an article about using the  to combine a number of resistors to achieve a specific value. Does anyone recall and have a copy of this article, or know how the Euclidean algorithm is applied to solve this problem? find a $\int_{x}^{g(x)} f(t)\,dt = 1$ Let $f : [0, 2] → \mathbb{R}$ be a continuous, positive function and  $$\int_{0}^1 f(x)\,dx = \int_{1}^2 f(x)\,dx= 1$$ For each $x \in [0, 1]$, prove that there exists a unique function $g(x) \in [1, 2]$ such that $$\int_{x}^{g(x)} f(t)\,dt = 1$$ and $g \in C^1$. Well, I think in $g(x) = x +1$, but I'm not sure. Any hint is appreciated.  Thank you. Why close and not migrate?