We, know every splitting field a finite algebraic extension. So, therefore every splitting field over the field with characteristics $0$ is a simple extension. But, here I choose the irreducible polynomial $x^{3}-2 $ in $\mathbb{Q}[x]$. Then , the splitting field of $x^{3}-2$ over $\mathbb{Q}$ is $\mathbb{Q}[\sqrt[3]{2},\sqrt{-3}]$ , But , now how could this extension $\mathbb{Q}[\sqrt[3]{2},\sqrt{-3}]$ generated only by one element? Also, how to show that Every finite extension of the field with characteristics $0$ is simple extension ?
Is there any elegant way to show that $\mathbb Q(\sqrt[3]{2}, w)=\mathbb Q(\sqrt[3]{2}+w)$, where $w=e^{i\frac{2\pi}{3}}$.  I was thinking to show that $ 9+9 x+3 x^3+6 x^4+3 x^5+x^6$ is the minimal polynomial of $\sqrt[3]{2}+w$, but checking irreducibility seems a bit tedious.
How do you evaluate this integral $\int \frac{6x^{3}+7x^2-12x+1}{\sqrt{x^2+4x+6}}dx$ $$\int \frac{6x^{3}+7x^2-12x+1}{\sqrt{x^2+4x+6}}dx$$  I have met this complex integral today, how do you evaluate this? Can you do this in more steps?  Is this an elliptic integral? I don't know how to evaluate this integral at all. Why does an argument similiar to 0.999...=1 show 999...=-1? I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the below argument that shows $999\ldots = -1$ is about? Here is the most usual proof I see that $0.999\ldots = 1$:  $x=0.999\ldots$ $10x=9.999\ldots$ $10x - x = 9$ $x=1$  Using this same argument template I can show $999\ldots=-1$:  $x= \ldots9999.0 $ $0.1x= \ldots9999.9$ $0.1x - x = 0.9$ $x=-1$  What might this mean? Edit from one of the comments: $$\sum_{k=0}^{\infty}{9 \cdot 10^k}=-1$$ Hot and Week tabs not shown to non-logged in users (except on SO) Visiting a SE site without logging in, one can no longer access the tabs Hot and Week (except on StackOverflow). I believe this change hides the feature from those most likely to use it: casual, but repeated, visitors. A number of people visit SE sites such as Travel or Mathematics just to read the content. They have no intention to post, and consequently, no reason to register. It makes little sense to hide the Hot and Week lists from them.   I propose that the tabs  ... Hot | Week | Month should again be made visible to anonymous visitors on SE sites, as they are on StackOverflow. I read something that suggests that if $I$ is the $n$-by-$n$ identity matrix, $v$ is an $n$-dimensional real column vector with $\|v\| = 1$ (standard Euclidean norm), and $t &gt; 0$, then $$\det(I + t v v^T) = 1 + t$$ Can anyone prove this or provide a reference? More generally, is there also an (easy) formula for calculating $\det(A + wv^T)$ for $v,w \in \mathbb{K}^{d \times 1}$ and some (invertible) matrix $A \in \Bbb{K}^{d \times d}$? Can somebody tell me why the counting measure (so, if $S=P(X)$, then $\mu(A)$=infinity if $A$ isn't finite and $\mu(A)=$#$A$ if $A$ is finite) is a measure? (The second property of a measure isn't clear for me). Simple Proof by induction: $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ I'm trying to prove using induction that $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ whenever $n$ is a non-negative integer. So far, I have:   Base case: $P(1) = (1) + (8) + (27) = 36, 36$ can be divided by $9$ so the base case is valid Inductive step: let $P(n)$ be the statement $9$ divides $n^3 + (n+1)^3 + (n+2)^3$. Assume $P(k)$ is true, so $9$ divides $k^3 + (k+1)^3 + (k+2)^3$.   And this is where I'm stuck. I'm not sure how to demonstrate that $9$ divides $P(n)$ when $n = k+1$. If someone could step me in the right direction that would be awesome. Thanks! Pivot in SQL 2008 R2 How to create a desktop shortcut in Unity? Is there an easy way to create a desktop shortcut for an application in Unity? The ability to right click or drag and drop icons to the desktop in Unity does not seem to be there. How do I run specific sudo commands without a password? Comprehensive list of tools that simplify the generation of LaTeX tables I have been building models with categorical data for a while now and when in this situation I basically default to using scikit-learn's LabelEncoder function to transform this data prior to building a model.  I understand the difference between OHE, LabelEncoder and DictVectorizor in terms of what they are doing to the data, but what is not clear to me is when you might choose to employ one technique over another.    Are there certain algorithms or situations in which one has advantages/disadvantages with respect to the others? Trigger to check for duplicate contacts on lead conversion I'd like to write a Lead trigger, to check for duplicate Contacts before inserting the lead, or converting it.   Contacts fields I'd like to check:   Firstname - should not be the same as the leads name Lastname - should not be the same as the leads name Email - should not be the same as the leads phone   After checking these fields &amp; failing to find a duplicate, only then should the lead be converted. I am trying to understand Maximum likelihood estimation but it looks like I am missing something rather elementary.          suppose we have an iid random sample $X_1, X_2,..., X_n$ for which the probability density function of each $X_i$ is $f(x_i; \theta)$ where $\theta$ is an unknown parameter. Then, the joint probability density function of $X_1, X_2,..., X_n$ is given by:          $f(X_1=x_1,X_2=x_2,\cdots,X_n=x_n)=\prod_{i=1}^{n} f(x_i, \theta)$      In Maximum Likelihood estimation, we try to maximize $f$ as a function of $\theta$.  Question: Why do we maximize probability density instead of probability? In what way does it make sense?  Edit: As explained in the answers, the probability of choosing a finite number of sample points from a continuous probability distribution is zero, so maximizing probability doesn't make sense, but how does maximizing probability density make sense? How can I identify what application is using a given file? I'm trying to delete some folder but this error is preventing from doing it:     Cannot delete [FileName]: It is being used by another person or program.      Close any programs that might be using the file and try again.   How can I identify what application is using the file?   I want to end the application using a Task Manager, but I didn't know what application is using it.  The folder is not shared and my user account is an administrator. Is it possible to disable X at boot time? I'm setting up a server so it would be nice if it wouldn't load the graphical interface every time I boot. Give the "some of your recent flags have been declined" banner some room The "some of your recent flags have been declined" message recently got a banner, which makes it stand out  more (which is nice) but it touches the 'Flag' button, which is less nice. Can it have some margin at the bottom (and perhaps less padding?)    Tested in the latest Safari and Firefox on macOS Catalina. Basis of cotangent space I just got this error message for the first time (I was getting this as the response to any request on the SE network for about the last five minutes, until now):     We're sorry...      There are an unusual number of requests coming from this IP address.   To protect our users, we can't process any more requests from this IP   address right now. We'll restore access as quickly as possible, so try   again soon. If you believe you have reached this page in error,   contact us.   As far as I know, I was not spamming SE or participating in DOS attack, and I find it unlikely that any of my office coworkers who share my IP address were doing so either.  Granted, I was pressing F5 a lot. But I've always pressed F5 a lot on SE, and have never gotten this message before. So my question is: Is it possible to get this error message simply by refreshing too much or too often? Or is that message reserved for a truly massive number of requests that are unlikely to be caused by one person refreshing a lot? (In the former case, I can't explain why I've never gotten it before; in the latter case, I can't explain why I just got it now). Fixed point, union of $\aleph_0, \aleph_{\aleph_0},\aleph_{\aleph_{\aleph_0}}\dots$ [see Noah's comment on notation] This actually triggered me in my mind from . After some playing around I notice that the relation $a^{\log_b x} = x^{\log_b a}$ is true for any valid value of $a,b$ and $x$. I am very inquisitive to see how this holds ?