Let $k$ be a field which is not algebraically closed and $n&gt;1$ be an integer . Then does there exist $f\in k[X_1,...,X_n]$ such that $Z(f) (:=\{(a_1,...,a_n)\in k^n : f(a_1,...,a_n)=0\}) =\{0\}$ ?  My Work : I was thinking like this : Considering $k$ as a sub-field of its algebraic closure $\bar k$, we have $k \subsetneq \bar k$ . Also, there is an irreducible polynomial $g(X) \in k[X]$ such that deg $g(X)&gt;1$ . May be we could somehow apply Hilbert Nullstelensatz to $\bar k$, but I am not sure .   I can easily do it for a finite field $k$; in that case just take  $f(X_1,...,X_n)=\sum_{(y_1,...,y_n)\in k^n \setminus \{0\} } \prod_{i=1}^n (1-(X_i-y_i)^{q-1})$ , where $|k|=q$ .  I can also do it if $[\bar k : k] $ is finite and $n=2$ . By Artin-Schreier theorem () , we will have $[\bar k : k]=2$ . Let $u\in \bar k \setminus k$, then the minimal polynomial of $u$ over $k$ has degree 2, and let $\bar u$ be another root of the minimal polynomial of $u$ over $k$ . Then $f(X_1,X_2)=(X_1+uX_2)(X_1+\bar u X_2)$ satisfies our required condition . Even in this case when $\bar k$ is a finite extension over $k$, I don't know what happens if $n&gt;2$ .  Please help . Thanks in advance
I am trying to prove the following statement.     If a field $K$ is not algebraically closed, then any $K$-variety $V\subset\mathbb{A}^n$ can be written as the zero set of a single polynomial in $K[X_1,X_2,...,X_n].$     Suppose $V=V(f_1,f_2,...,f_n)$, and I would want to find a polynomial $g\in K[X_1,X_2,...,X_n]$ such that $V=V(g)$.  I know that if I can find a polynomial $\phi\in K[X_1,X_2,...,X_n]$ whose only zero is $(0,0,...,0)$, then it is done, since we could take $g=\phi(f_1,f_2,...,f_n)$.  For $n=1$, then I would just choose $\phi=X$, but I do not even know how to proceed to the case $n=2$. Maybe we could choose an irreducible polynomial of degree $&gt;1$.  Any help will be appreciated.
I have been reading Stack Exchange for years - mostly not logged in, and have never posted (this is my first). I'm simply not very talkative, and very good at googling.  Recently a few occasions have come up when I've wanted to comment on a question.  I wanted either to ask for, or provide, clarification which did not warrant a new answer.  How do I gain enough reputation to comment without:   asking questions which have already been answered or that I can figure out on my own, or posting my comment like as an answer (i.e. being  rude)? Answer textbox's content starts overlapping when adding new lines in middle on Android app When typing/editing an answer, sometimes if I scroll the editor textbox, go upwards and add lines, the text will start overlapping each other. The behaviour is that when you add a newline, the text below it does not move downwards.  Minimizing soft keyboard fixes this.  This does not affect the Ask Question box in my test.    I am using Android 4.1.2 Is there a way to bypass resistance/immunity to fire damage? I want to use fire-based spells for thematic reasons and a lot of things have immunity to it.   Is there a feat or class feature I could use to bypass/reduce those? For example, is there something that ignores fire resistance and reduces fire immunity to just resistance?   I know the easy way around it is just to pick another damage type spell but I really want to roleplay a pyromancer/firebender-type character. I was on the tube and overheard a dad questioning his kids about maths.  The children were probably about 11 or 12 years old.   After several more mundane questions he asked his daughter what $1/0$ evaluated to. She stated that it had no answer. He asked who told her that and she said her teacher. He then stated that her teacher had "taught it wrong" and it was actually $∞$.  I thought the Dad's statement was a little irresponsible.  Does that seem like reasonable attitude? I suppose this question is partly about morality. "physical block size is 2048 bytes, but Linux says it is 512" when formatting USB - How to create a bootable USB without this error Is there a way to make bash more TAB friendly? I use bash on both my Mac and as my cygwin shell. Overall, I'm quite pleased with it.  My one complaint however is that its TAB completion isn't as good as Windows Command Line TAB completion.   For example, if there are two files with the same prefix, then bash just beeps at me when I try to TAB complete. If I keep nailing TAB, then it will eventually show me a list of potential files.  I definately prefer how Windows will cycle through potential files right on the command line.   Is there a way to get bash to operate similar to Windows Command Line in this regard?  If there isn't (I do not want to switch shells), am I missing something about TAB completion in bash? Is there a more efficient way to use it? Nested roots sequence, how to prove it's monotone and bounded? Let $a\ge1$ and define the sequence $(x_n)$ recursively by: $$x_1 = \sqrt{a}$$ $$x_{n+1}= \sqrt{a+x_n}$$  Here's what I did:  Plugging in some values makes it seem as if the sequence is increasing. I first solve the equation $t=\sqrt{a+t}$, equivalently $t^2 -t -a =0$, to get an idea what's going on. The roots are $$t_{1,2}= \frac{1 \pm \sqrt{1+4a} }{2}.$$ Therefore, between the roots, that is for $$t\in \left( \frac{1 - \sqrt{1+4a} }{2}, \frac{1 + \sqrt{1+4a} }{2}\right)$$  we have $t^2-t-a \le 0 \iff t \le \sqrt{a+t}$. Here's my attempt at showing inductively that $$\forall n\in \mathbb{N}, \ x_n \le \frac{1 + \sqrt{1+4a} }{2}$$  For $n=1$, the statement is true (simple arithmetic). Suppose it's true for $n$. Consider  $$x_{n+1}= \sqrt{a+x_n}\le \sqrt{a+\frac{1 + \sqrt{1+4a} }{2}}=\frac{1 + \sqrt{1+4a} }{2},$$  because $\frac{1 + \sqrt{1+4a} }{2}$ satisfies $x=\sqrt{x+a}$.  So this establishes the sequence $(x_n)$ is bounded. Now, by induction again, I try to show it's monotonically increasing. The base case $x_1\le x_2$ is obvious. Assume (strong induction) $x_{n-1}\le x_n$ for all $n&gt;1$. But since we have $$x_1\le x_n \le \frac{1 + \sqrt{1+4a} }{2}, $$   then $x_n$ lies in the appropriate interval, so $x_n \le \sqrt{a+x_n}=x_{n+1}.$  Is this proof correct? Can the condition on $a$ be relaxed to $a&gt;0$? Maybe even small negative $a$? And could someone provide me with another proof of convergence of $x_n$? I absolutely hate this one! Proving that orthogonal vectors are linearly independent I am trying to prove that $a_1$, $a_2$, $a_3$ are linearly independent.  I am asked to use vector product and prove that if $c_{1}a_{1} + c_{2}a_{2} + c_{3}a_{3} = 0$ then $c_1 = c_2 = c_3 = 0$  I am completely stuck on where to go with this problem. I would think that linearly independent then the null space of the space can only be equal to $0$. Would this prove it? This question is essentially the same as one of my , except the only difference is that I am using "only" to change the text at the same location (i.e., ). In other words, I am destroying my earlier text as I go, and not adding new ones underneath the previous ones.   Since the text for the first "only" is shorter than the second "only", therefore the image will shift location. I would like for the image to remain at its location.   Here is an example     it may not be obvious but the second image is higher up than the first.  The code is here, can someone find a solution?  \documentclass{beamer}  \usepackage{graphicx}  \title[Title]{Presentation} \author{Sandro Botticelli} \institute{Italy} \date{1484}  \begin{document}   \begin{frame} \begin{figure} \includegraphics[scale = 0.8]{Venus}     \caption{Nascita di Venere} \end{figure}  \only&lt;1&gt;{In the centre the newly-born goddess Venus stands nude in a giant scallop shell. Its size is purely imaginary, and is also found in classical depictions of the subject. }    \only&lt;2&gt;{Alternative identifications for the two secondary female figures involve those also found in the Primavera; the nymph held by Zephyr may be Chloris, a flower nymph he married in some versions of her story, and the figure on land may be Flora. Flora is generally the Roman equivalent of the Greek Chloris; in the Primavera Chloris is transformed into the figure of Flora next to her, following Ovid's Fasti, but it is hard to see that such a transformation is envisaged here.} \end{frame}   \end{document} Dublin transit visa requirements for Indian citizens How to randomly subset X% of selected points? What methods are available in ArcGIS 10.2 to randomly subset a selection of points.  For example, in the attached screenshot I am interested in keeping 20% of the selected points and deleting the rest. hot network questions have inconsistent CSS for visited/not visited depending on the site you're on This is a HNQ excerpt from Stack Overflow. Visited questions are objectively darker.    This is a HNQ excerpt from Super User. Visited questions are objectively lighter.    This is a HNQ excerpt from Seasoned Advice. Visited questions are objectively darker.    This is a HNQ excerpt from Puzzling, a beta site. Visited questions are objectively darker, although the difference is hard to spot.    This is a HNQ excerpt from Travel. Visited questions actually have a completely different color.    These are 5 different sites, with 5 different formats for how the HNQ sidebar displays links. Some make visited links darker. Some make them lighter. Some make them a little big darker. some give visited links a completely different color. This is very confusing: If someone visits one of these HNQ questions and wants to continue browsing the HNQ from there, they likely expect that the links in that HNQ segment have consistent layout compared to the HNQ segment from the website they came from.  Is this something worth looking into? I realize that each site has their own look and feel, but an element that appears on every site should be consistent IMHO. People have expectations about how links work, and one of those is that if you click on a link within the same domain (which the HNQ segment arguably is), they behave similary on that page as they do on the page you come from. How to add a directory to the PATH? What is the sponsored party called? Google results for sponsee have scattered definitions in unreliable sources, so it does not appear that is a real word. I'm looking for something similar, a single word.  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The following quote from  assumes "the URL" comes from somewhere and uses it to restrict the Host header. Where does "the URL" come from? Is it from the first line of a HTTP request (e.g. after GET), or from the input to a web client (e.g. an argument to curl, or the input in the address bar of a web browser)? Thanks.     The Host header specifies the Internet host and port number for the   resource being requested, as obtained from the original URL:  Host = "Host" ":" host [ ":" port ]       In particular:      • If the Host header does not contain a port, the default port for the   scheme is assumed.      • If the URL contains an IP address, the Host header should contain   the same address.      • If the URL contains a hostname, the Host header must contain the   same name.      • If the URL contains a hostname, the Host header should not contain   the IP address equivalent to the URL’s hostname, because this will   break virtually hosted servers, which layer multiple virtual sites   over a single IP address.      • If the URL contains a hostname, the Host header should not contain   another alias for this hostname, because this also will break   virtually hosted servers. How to set security options for a PDF using pdflatex My questions is very simple. 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