Does anyone know where I can find a good proof for the basis theorem of the cohomology ring of the Grassmannian, or give me a sketch of the proof?  I'm already familiar with basic Schubert Calculus.
Let $G(r, n)$ be the Grassmannian of the set of all $r$-planes in a $n$-dim vector space. How to show that the cohomology of a Grassmannian has a basis consisting of the equivalent classes represented by Schubert cycles? I am confused since I don't know how to compute the cohomology of a variety (how to construct the co-chain complex). I am really appreciate if you can compute the cohomology of, for example, $G(2,4)$ and show that it has a basis consisting of the equivalent classes represented by Schubert cycles.
Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$? We know that $f(x)=x^4+1$ is a .  My question is:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ has a linear factor over $\mathbb{Q}$?   Edit: Thanks for @Jyrki Lahtonen's answer, I want to do some modifications:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?   Thanks in advance! I accidentally opened Orca Screen Reader from the Dash when I was trying to open Screenshot. Now everything I type or click on is spoken out loud.  How do I make it stop?  For reference, here is a screenshot of Orca Screen Reader running: Custom Post Type Archives by Year & Month? How do you display the archives of a Custom Post Type by Year &amp; Month? Why does the dock appear on the left when my screen is locked? The dock doesn't work correctly after I enter my password, either. I have already read the question about  but this is not what I want, I want to be able to click on the file in Dash or in Nautilus and be able to open it with the specified app.  For example I have this case right now:  When I click on an EXE file it does not get executed by Wine, instead I get the following:    If I try to look for Wine in the Open With context menu option it is there. I am fine executing it from there the first time, but then the following times Wine does not appears selected in the main Context menu:    I then went to Properties to the Open With Tab to look for Wine. It was not in the Default apps nor was it in the Recommended ones when I clicked and search on Show other applications    This is not limited to EXE extensions. It also has happened with MKV files, MP4 files, JPG, PNG, MP3 and even SVG and CSS. So instead of trying to find a way of solving it for one, I want a way that can be done to any type of extension that can:   Make the app the default of that extension Appear on the context menu in Nautilus when I right click on the file Appears on the "Open With" tab on the Property window of the file Correctly executes the specified app when I click on the file with Dash   This might sound like it should be common sense for an app to execute the file but it does not always work like that. So for the cases where it actually does not work, how to fix it. A reset of the extension?, An update of the files that can open said extension?, an addition of the app via a command line or GUI tool? Include the option to flag posts as spam in the Low Quality queue I guess this is an additional to previous requests like  and .  But I just came across  review item, and was unable to flag it as spam.  edit: Since I posted this, the item has been removed (deleted) here is a screenshot: How do we see? Where do the photons disappear? I know that the light is reflected from a object to my eyes, but I don't understand exactly how. The photons appear from the light source and disappear in my eye! Can someone explain the phenomenon of where the photons go and do to allow us to see? Which of the following polynomials are subspaces of $\mathbb{P}_n$ for an appropriate value of n? Definition. A subspace  of a vector space V is a subset H of V that has three properties  (a) The zero vector of V is in H.  (b) $\forall$ u, v $\in H$, we have  u + v $\in H$.   (c) $\forall$ u $\in H$, c scalar, we have c  u $\in H$.  For $n \geq$ 0, denote $\mathbb{P}_n$ to be the set of all polynomials of degree at most $n$.  Which of the following are subspaces of $\mathbb{P}_n$ for an appropriate value of $n$? Support your answer with reason.  (a) All polynomials of the form $p(t) = a + t^2$, where $a \in \mathbb{R}$.  (b) All polynomials of degree at most 3, with integers as coefficients.  Solution:  Both the statements (a) and (b) above are not subspaces of $\mathbb{P}_n$.  Reason:  For part (a), All polynomials of the form $p(t) = a + t^2$, where $a \in \mathbb{R}$, is not a subspace because it failed to meet the standards of the first property stated above, that is, the coefficient of the $t^2$ term is always a 1, therefore the zero vector is not in this subspace of polynomials.    For part (b), All polynomials of degree at most 3, with integers as coefficents is also not a subspace becuase it as well failed to meet the standards of the third property, that is, integer coefficents, when multiplied by a real scalar, will not necessarily be of integer type leading to integer coefficents. Thusly, the resulting polynomial will not be in the subset of polynomials $\mathbb{P}_n$. $\Box$  Can someone please look at this and see if I have the right idea and approach to this question. Thank You Lost passport with Schengen visa boolean variable values in PHP to javascript implementation Conditional Expectaction 3 variables Suppose $X,Y$ and $Z$ are multivariate normal with means and full covariance matrix. The conditional expectation $E(X | Y)$ is well know. What is the conditional expectation of $E(X | Y,Z)$ if $Y$ and $Z$ (and $X$) are correlated? Standard textbooks only seem to cover the case when $Y$ and $Z$ are uncorrelated. Graphical glitches using Ubuntu 18.04 on Dell XPS 13 2019 laptop My laptop keeps on having weird screen glitches, sometimes for a second, and that goes away just by moving the cursor, sometimes, making me restart the whole thing.  I could still use the play/pause key and it still played/paused the video that was running even though the screen was completely teared.  Also, printscreens actually print a normal screen and do not show the problem.  Some pictures of the screen when it happens :     Do you guys know what I could do to solve the issue ? How many different (circular) garlands can be made using $3$ white flowers and $6m$ red flowers? This is my first question here.   I'm given $3$ white flowers and $6m$ red flowers, for some $m \in \mathbb{N}$. I want to make a circular garland using all of the flowers. Two garlands are considered the same if they can be rotated to be the same. My question is: how many different garlands can I make?  I tried an approach in which I divided the question into three parts:   No. of ways in which the white flowers are all together No. of ways in which two white flowers are together No. of ways in which no two white flowers are together   Can you please help?  The answer is $3m^2+3m+1$ in the book that I saw  Thanks Is there a word for the sadness over "What might have been"? Context: You made the decision not to see a person any more. The relationship was good, you were both happy with one another but the one main stumbling block (for you) in the relationship was never dealt with. It remained, a permanent fixture, that forced you to make that sad, and final decision.   Is there a word, an idiom, or a famous line that expresses the sadness for something that might have been; e.g. a rich and fulfilling relationship. It's not nostalgia, because that expresses a wistfulness for the past.   Instead, you are wistful for the future. 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. Is the EmDrive, or "Relativity Drive" possible? How do I replace the *first instance* of a string in .NET? The computed moments of log normal distribution can be found . How to compute them? There's an election going on. What's happening and how does it work? What is the name of this Ubuntu program? A couple month ago I installed a bunch of clipboard manager programs in Ubuntu, tried them out and then removed them. I must have missed one of these and now it keeps bug me in my daily usage. I need some help to identify the name of this program because the program itself provides very limited information.  This clipboard program has below characteristics: 1. It occupies hotkey Ctrl+;. When I press this hotkey, the clipboard history will pop out. It happens in bash, Chrome, Eclipse IDE, and etc, wherever it is text inputable. This is kind of annoying because it overrides my other hotkeys inside these applications. 2. Very laconic UI. Like below picture, with only a line "Select to paste", then listing out the clipboard history. There is no program information whatsoever. Right clicking on it and no menu are shown at all.      So far I have tried below ways to identify this program: 1. Browse system monitor for processes with names that could be correlated. 2. Browse through my installation log (dpkg.log) to find my installation history. Without luck, the log is lost. But I was not successful.