For which ideals $I$ is $k[x_1,\ldots,x_n]/I$ finite degree extension of $k$?
Zero-dimensional ideals and finite-dimensional algebras
Upload files with HTTPWebrequest (multipart/form-data) Let $d=GCD(n^2+5,n^3-5n^2+6n)$ show that $d|630$ How to add startup applications in lubuntu? Is there an intuitive explanation for why Lorentz force is perpendicular to a particle's velocity and the magnetic field? Have Google index different languages at the same URL instead of just the default (English) I have a multilingual PHP site. I have set English as the default language.  The language changes automatically according to the browser settings. If it is French, the site will be seen in French.  However in the Google search results, even if I have the browser in French, the English language appears in the results. How can I fix that?  To be clear I don't want to use subdomains or URLs like www.example.com/fr/. I would like it to work like Facebook. In the film and book series, Year 4 (Goblet of Fire), how is it Mad-Eye Moody/Barty Crouch could see Harry through his Magical Eye even when he was under the Invisibility Cloak?  As the cloak is a True Invisibility cloak, immune to charms and spells, and according to legend invisible to even Death, how did Mad-Eye detect Harry Potter under the cloak? How Nagini and Mrs. Norris were able to sense Harry were through other senses, such as smell and warmth, but not visually.  I'm referring to the scene/chapter where Snape follows Harry at night. Harry hides under the Invisibility Cloak but his foot is stuck at the broken staircase. Eventually Mad-Eye arrives, sees Harry through his Magical Eye, and talks Snape away. Two  have been announced in LoL.  They have an ability that makes them be able to recall together.                                  What are the limits to this ability? Can the 'non-recaller' jump in at the last second and get some kind of instant recall?  Can somebody with the setup to get on PBE test it and tell us what are the limitations of this ability? Creating or referencing variables dynamically in Sass This is a follow up question to my previous question  Let $k$ be a field and $V \subseteq \Bbb{A}^n$ and $W \subseteq \Bbb{A}^m$ be algebraic sets. Then it should be true that $I(V \times W ) = I(V) + I(W)$ where by $I(V)$ here we mean the extension of $I(V)k[x_1,\ldots,x_{m+n}]$. Now I believe I have proven this (see the proof at the bottom of my question) but when I look at Martin's answer , it is instead claimed that we  actually have  $$I(V \times W )= \sqrt{I(V) + I(W)},$$  and for $I(V) + I(W)$ to be a radical ideal we need $k$ to be algebraically closed.           My question is: What's going on here? I believe my claim is true even without the assumption that $k$ is algebraically closed.           Here's a proof user Sanchez told me of, which I have simplified:       First it is clear that we always have $I(V) + I(W) \subseteq I(V \times W)$. For the reverse inclusion consider a polynomial $f \in I(V \times W)$. Then we can always write $$f = \sum_{i=1}^n f_ig_i$$ where $f_i \in k[x_1,\ldots,x_n]$ and $g_i \in k[x_{n+1},\ldots,x_{m+n}]$. Now take any $b' \in W$. If for all $i$ we have $g_i(b') = 0$ then since $b'$ is arbitrary, $g_i \in I(W)$ for all $i$. Then $f \in I(W) \subseteq I(V ) + I(w)$ and we are done. Otherwise suppose there exists $b \in W$  and $i$ such that $g_i(b) \neq 0$. Then wlog we may suppose that  $g_1(b) \neq 0$.   Next, $\sum f_ig_i(b) = 0 $ on all of $V$ by assumption of $f \in I(V \times W)$. So $\sum f_i g_i(b) = p$ for some $p \in I(V)$. Now write $$f_1 = \frac{ p - g_2(b) f_2 + \ldots g_n(b)f_n}{g_1(b)}.$$  Substituting this for $f_1$ in $\sum f_ig_i$, followed by taking things mod $I(V)$ we get an  expression with only $n-1$ terms $\mod{I(V)}$. Continuing this process we will finally get an expression with $0$ terms $\mod{I(V)}$ so that $f \in I(V)$. This shows $I(V \times W) \subseteq I(V) + I(W)$ which completes the proof. Pointer to class data member "::*" I came across this strange code snippet which compiles fine:  class Car {     public:     int speed; };  int main() {     int Car::*pSpeed = &amp;Car::speed;     return 0; }   Why does C++ have this pointer to a non-static data member of a class? What is the use of this strange pointer in real code? Have we ever observed two drosophila lineages that evolved reproductive isolation in labs? Background  The standard definition of species refers to the concept of . If two lineages are found to be reproductively isolated, then we consider these two lineages to belong to different species. My question concerns evolved reproductive isolation in Drosophila sp. following labs due to experimental evolution.  Question  Have we ever demonstrated that two Drosophila sp. lineages that could initially interbreed (in nature or in labs) evolved through artificial selection (and drift and mutations) in labs to finally not be able to interbreed anymore either due pre- or post- zygotic isolation (see )? Or, in other words, have we ever demonstrated that two drosophila lineages evolved to become different species (reproductive isolation definition) in labs experiments?  If not, have we ever observed some partial reproductive isolation such as inbreeding depression for example?    Note: This question is motivated by @LotusBiology that could not receive the answers she/he was waiting for because he/she failed to ask questions that are possibly answerable! So I wanted to ask this question that somehow addresses  question he/she asked (now on-hold)  For more information about the concept of species, please have a look at How to repair a broken .EXE file association How to store credit card info on iphone? Inexact measurement and wavefunction collapse Sphere intersection Stuck I have two spheres that are bounded by spheres for collision detection. I do not want them to intersect. My intersection works. But it gets stuck. Once the function returns true, I cannot get it out. I tried  collideBool = ObjectOne.Intersect(ObjectTwo);  if(collide == true) {     leftright = leftright;     updown = updown; } else { if(GetAsyncKeyState(VK_RIGHT)) leftright -= dt; if(GetAsyncKeyState(VK_LEFT)) leftright += dt; if(GetAsyncKeyState(VK_DOWN)) updown += dt; if(GetAsyncKeyState(VK_UP)) updown -= dt; }  collide = false;   My sphere gets stuck. What is a way around this? Simplify a combinatorial sum Why is $\sum_{k=0}^n (_k^n) 2^k$ simplified as $3^n$ and not just $2^n$ ? The answer uses the binomial theorem as a solution: let $a=1$ and let $b=2$. So $1+2 = 3$. But since $1+2 = 3$ why do we not use $1+1 = 2$? Can you be refused a visitor visa to US if you have bad Credit there? I did my 3 years of College in the US, 2010-2013 due to finance issues I was unable to pay my credit card bill of a certain bank, which still lie unpaid.  However I plan on visiting the US later this year; could I be denied visa for having an unpaid bill ? Is $\frac{1}{\sqrt{2}}$ a fraction or not?   I just came across a Grade 8 student's class work where the teacher has given the above as an example to differentiate fractions from rational numbers. I think this is wrong fundamentally, as $\sqrt2$ is irrational while all fractions would be rational. Please advise. Thanks! How do I put a backdrop behind the model that I am creating so that I can compare the picture to the model while editing? If $R$ is an integral domain and a subring $S$ has identity $1_S$, how would you show that $1_S=1_R$ (here $1_R$ is the identity of the ring $R$)? I am unsure about what an integral domain really is and how the subring comes into play here.