Let $p_1,...,p_n\in\Bbb N$ be distinct primes. Compute $\text{Gal}(\Bbb Q(\sqrt{p_1},...,\sqrt{p_n})/\Bbb Q)$  Intuitively, $G:=\text{Gal}(\Bbb Q(\sqrt{p_1},...,\sqrt{p_n})/\Bbb Q) = (\Bbb Z/2\Bbb Z)^n$. My first idea is that if $\sigma\in G$, then $\sigma(\sqrt{p_i})=\{\sqrt{p_i},-\sqrt{p_i}\}$ for each $i$. Hence, from this, pick $\sigma_i\in G$ such that $\sigma_i(\sqrt{p_i})= -\sqrt{p_i}$ and $\sigma_i(\sqrt{p_j}) = \sqrt{p_j}$ for all $j\neq i$. Then, $\sigma_i$'s generates $G$ and each $\sigma_i$ does not generates $\sigma_j$ if $i\neq j$, $G = (\Bbb Z/2\Bbb Z)^n$. Is this valid? Another way to do this is using the fact that 'if $K/F,L/F$ are both Galois and $K\cap L =F$ then $\text{Gal}(KL/F)\simeq\text{Gal}(K/F)\times\text{Gal}(L/F)$', we inductively apply this. The problem is whether $\sqrt{p_i}\in\Bbb Q(\sqrt{p_1},...,\sqrt{p_{i-1}})$. Any hint or comment will be appreciated.
For distinct prime numbers $p_1,...,p_n$, what is the Galois group of $(x^2-p_1)\cdots(x^2-p_n)$ over $\mathbb{Q}$?     This problem appears to be quite common, however my understanding of Galois theory is quite poor, and I have no idea how to do this problem.
Missing settings in GNOME control center installed in Ubuntu MATE I installed GNOME Control Center following  to access Google Drive in Ubuntu MATE 16.04 LTS (MATE desktop environment 1.12.1)   The problem is GNOME Control Center is only showing language support, printers, firewall setup and backups... not online accounts or other settings.  This is what I get:  ~$ gnome-control-center gnome-online-accounts   ** (gnome-control-center:5744): WARNING **: Ignoring launcher landscape-client-settings (missing desktop file)  ** (gnome-control-center:5744): WARNING **: Invalid categories System;Settings; for panel software-properties-gtk.desktop  ** (gnome-control-center:5744): WARNING **: Ignoring launcher ubuntuone-installer (missing desktop file)  ** (gnome-control-center:5744): WARNING **: Could not find settings panel "gnome-online-accounts"   Any ideas?  Thanks! Make an Ubuntu bootable USB drive for a PC using Mac? How to measure time taken by a function to execute How do I kill background processes / jobs when my shell script exits? I am looking for a way to clean up the mess when my top-level script exits.  Especially if I want to use set -e, I wish the background process would die when the script exits. Restored iOS 10 with broken Home button I restarted my iPhone 6 running iOS 10 and the home button is broken, so I was using Assistive Touch. But at the first launch, Assistive Touch is disabled. On iOS 10, we have to "Press home button to unlock".  What can I do? How to know the usage of undocumented function like LinearAlgebra`BLAS? How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$? Show that $\,\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$.   I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to $i!/2!(i-2)!$ and simplify from there? Not sure if that will help, not to mention if I put $1$ in for $i$ I get $1/-2$ which I don't think is right.   Anyone care to shed some light on the subject?   Thanks. Are confidence intervals open or closed intervals? First chapter appearing before first line in table of contents Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ Let $X=C[0,1]$ be the set of all continuous functions on $[0,1]$. For any two functions $f,g\in X$, set $$d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|.$$  Prove $(X,d)$ is a metric space.    My attempt:  This question comes down to showing that $d$ is a metric, which means that:   $d(f,g)\geq 0$ $d(f,g)=0\iff f=g$ $d(f,g)=d(g,f)$ Triangular sadness.     1) Nonnegativity is guaranteed by taking the supremum of the absolute value of a sum, since the absolute value of a sum is non-negative, the supremum of non-negative numbers is non-negative.  $\square$    2) For the supremum of $|f(x)-g(x)|$ to be equal to $0$, the greatest number for $|f(x)-g(x)|$ from $0\leq x\leq 1$ must be $0$, hence $|f(x)-g(x)|=0$ for all values on this range, and thus $f(x)=g(x)$ is the constant zero map.  $\square$    3) $|f(x)-g(x)|=|g(x)-f(x)|$ implies that $\sup \limits_{0\leq x \leq 1} |f(x)-g(x)|=\sup \limits_{0\leq x \leq 1} |g(x)-f(x)|$ $\square$    4) Triangular sadness:  We need to prove that $d(f,g)\leq d(f,h) + d(g,h)$  I.e.  $\sup \limits_{0\leq x \leq 1} |f(x)-g(x)|\leq \sup \limits_{0\leq x \leq 1} |f(x)-h(x)|+\sup \limits_{0\leq x \leq 1} |g(x)-h(x)|$  Now I am not sure how I can 'behave' with the sup 'function'. Can I do the following?  $$\sup|f(x)-g(x)| = \sup|f(x)-g(x)+h(x)-h(x)|=\sup|f(x)-h(x)+(h(x)-g(x))|$$ Then use normal triangular inequality:  $$\sup|f(x)-h(x)+(h(x)-g(x))|\leq \sup(|f(x)-h(x)|+|h(x)-g(x)|)=\sup(|f(x)-h(x)|+|g(x)-h(x)|)$$   This is obviously less than $\sup \limits_{0\leq x \leq 1} |f(x)-h(x)|+\sup \limits_{0\leq x \leq 1} |g(x)-h(x)|$, but I don't know how to prove it. How do I finish that one off. How do I remove 'Places' entries from the Nautilus sidebar? I want to remove Videos and Music from the left side panel. But, I can't seem to find the option to do so. How do you do that? How can you profile a Python script? Do I need the original discs to install WoW on a new computer? My boyfriend bought the World of Warcraft Battle Chest, Wrath of the Lich King, &amp; Cataclysm. He downloaded them to our previous computer, but now we have a new one and when he took out the first disk from the battle chest he severely scratched it. Anyway, I tried putting in disk #2, and it said I couldn't do it without disk 1 (which is scratched!).  I've recently put in Wrath of the Lich King (it came with 2 CDs), and it said that I haven't downloaded the original and asked me if I wanted to downloaded. I clicked "yes" -- does that mean its downloading WoW &amp; The Burning Crusade? Or do we need to buy it all again?  We just bought the whole set on June of this year... If $p$ is congruent to $2 \pmod 3$, how can I prove that all $a$, $1 \le a \le p-1$ are cubic residues $\mod p$? If $p$ is congruent to $2 \pmod 3$, how can I prove that all $a$, $1 \le a \le p-1$ are cubic residues $\mod p$?  Here's what I've done:  $1^3$ congruent to $1 \pmod p$ thus, 1 is a cubic residue, Also $(p-1)^3=(p-1) \pmod p$ implies  $$(p-1)^3 - (p-1) = (p-1)[(p-1)^2 -1] = (p-1)(p^2 -2p) = p(p-1)(p-2) $$ implies $$p|(p-1)^3 -(p-1)$$ Thus  $p-1$ is a cubic residue $\mod p$. Now I don't know how to show for $1\lt a \lt p-1$ are all cubic residues when $p$ is congruent to $2\pmod 3$. Why commutator subgroup is normal to G? If $G$ is a group and $G'$ is generated by $\{xyx^{-1}y^{-1}|x,y\in G\}$, then $G'\trianglelefteq G$ and  $G/G'$ is Abelian.  At first, I thought this is easy because I thought $g[x,y]g^{-1}=[gxg^{-1},gyg^{-1}]$. But it was a mistake. I could not prove that it is also subgroup of G and $G/G'$ is abelian. I need help. Problems updating Python from the repositories today (Feb 25, 2021) NOTE: This is a Master Canonical Post for the issues observed on February 25, 2020.  All issues on Feb 25 relating to Python updates not found should be dupe-linked here.  Errors have been seen such as this today, February 25, 2021: Failed to fetch http://security.ubuntu.com/ubuntu/pool/main/p/python2.7/libpython2.7-minimal_2.7.17-1~18.04ubuntu1.3_amd64.deb 404 Not Found  Is there something up with the repositories? some more examples:   "And she gave me that aren't-I-just-gorgeous smile." "The I-did-it-my-way approach." "A from-this-day-forward-I-have-no-son scene." The add comment link and comment box overlaps bounty box I am looking for a convenient way to set the background of tabular cells. I'd like to do two things:  (1) Set the background shaded in a color (2) Set the background hatched  I learned that the TikZ package provides means to do this but I am unable to come up with a solution that does not use matrices (such as here: ) but instead uses the tabular package.  Here's what I'd like to do:  \documentclass{article} \begin{document} \begin{tabular}{|l|l|l|} \hline   1 &amp; \setBGGray 2 &amp; 3 \\\hline   4 &amp; 5 &amp; 6 \\\hline   7 &amp; \setBGHatched 8 &amp; 9 \\\hline \end{tabular} \end{document}   The output should look like this:    Can someone please help me achieving this using tabular (or tabularx) and TikZ? Why are high white balance temperatures redder when warmer objects are bluer? I know it's dumb to ask this type of questions. But, I still don't get this. I've read the "Black body" story and know that when it's heated, it gives off colors; 'warm' colors (red→orange→yellow..) that have low color temperatures, at low temperatures, (I mean, when it is cooler than afterwards) right? And, the black body gives off 'cold' colors (white→blue..) that have high color temperatures at high temperatures (when it is hotter than before), right? Correct? So there're two, yes, very dumb, questions and the latter one is dumber..     Why are that 'warm' colors said to have low temperatures? And 'cold' colors vice versa? So the redder colors have lower c/t and bluer colors have higher c/t, then   why does the camera's color temperature - adjusting slider has the bluer colors on the left, and the redders on the right?       Please, forgive me for my ignorance...