Show that $\mathbb{Q}(\sqrt{a} + \sqrt{b}) = \mathbb{Q}(\sqrt{a}, \sqrt{b})$ Let $a$ and $b$ be rational numbers. Show that $\mathbb{Q}(\sqrt{a}, \sqrt{b}) = \mathbb{Q}(\sqrt{a}+ \sqrt{b})$.  I think this question might have been asked before, but Google and the stackexchange search don't show anything.
Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}), \forall a,b\in\mathbb{Q}$ Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then   $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}),  \forall a,b\in\mathbb{Q}$   It has something to do with $\sqrt{a} = -\sqrt{b}\implies a^2 = b^2$  The inclusion $\mathbb{Q}(\sqrt{a}+\sqrt{b})\subseteq \mathbb{Q}(\sqrt{a},\sqrt{b})$ is obvious  Now take $x\in \mathbb{Q}(\sqrt{a},\sqrt{b})$, that is: $x = c + d\sqrt{a} + e\sqrt{b} + f\sqrt{ab}$. I need to show that this is in $\mathbb{Q}(\sqrt{a}+\sqrt{b})$, that is, $x = g + h\sqrt{a} + i\sqrt{b}$, but I can't get that $\sqrt{ab}$ removed
Number of possible combinations of x numbers that sum to y I want to find out the number of possible combinations of $x$ numbers that sum to $y$. For example, I want to calculate all combination of 5 numbers, which their sum equals to 10.  An asymptotic approixmation is also useful. This question seems to be very close to number partitioning, with the difference that a number can be 0. See:    All possible partitions for sum 10 and 3 positions that can be zero, are 63 possiblities: (numbers shown as 3 digits)  019 028 037 046 055 064 073 082 091 109 118 127 136 145 154 163 172 181 190 208 217 226 235 244 253 262 271 280 307 316 325 334 343 352 361 370 406 415 424 433 442 451 460 505 514 523 532 541 550 604 613 622 631 640 703 712 721 730 802 811 820 901 910 Setting nautilus-open-terminal to launch Terminator rather than gnome-terminal Systemd backed tmpfs | How to specify /tmp size manually I mount /tmp on tmpfs using:  sudo systemctl enable tmp.mount sudo systemctl start tmp.mount   But this way /tmp takes up all the free RAM:  $ df -h /tmp Filesystem      Size  Used Avail Use% Mounted on tmpfs           3.9G   12K  3.9G   1% /tmp $   How do I tell systemd tmp.mount to use only 1G? I know I can alternatively not use systemd and manually add an entry to /etc/fstab and specify the size there. But I don't want to do that. I want to use systemd backed tmpfs. How to reconnect a logically disconnected USB device? It is possible for a USB storage device to become logically disconnected: the device is still plugged in, but is invisible from the operating system (e.g. it's not listed under /proc/bus/usb). Maybe this can happen to other kinds of USB or other removable devices as well. This can happen for example:   ; after VirtualBox has claimed a USB storage device (the device disappears from the Linux host when it is attached in the guest, and does not automatically reappear if it is detached from the guest).   What is going on in the driver or in the hardware, and most importantly, (how) can the device be logically reconnected (without physically plugging it out and back in)?  Note that this question is not about mounting. In the “logically disconnected” state, the kernel believes there is nothing on the USB bus (and so of course there is no entry under /dev). I am trying to figure out this:  c = 1 def f(n):     print c + n  def g(n):     c = c + n  f(1) # =&gt; 2 g(1) # =&gt; UnboundLocalError: local variable 'c' referenced before assignment   Thanks! $\lambda=\frac{2h}{p}$ instead of $\lambda=\frac{h}{p}$? I am studying quantum physics and there is something I don't understand:   I know that for any particle $E=hf$ (Einstein relation) and  $v=\lambda f$ ($v$ is the speed of the particle). I also know that the kinetic energy is $E_k=\frac{mv^2}{2}$.    Solving those 3 equations for $\lambda$:  $$h\frac{v}{\lambda}=\frac{mv^2}{2},$$ I finally find $$\lambda=\frac{2h}{mv}=\frac{2h}{p},$$ which is not consistent with the De Broglie relation $$\lambda=\frac{h}{p}.$$ Where am I wrong in my development? Setting Objects to Null/Nothing after use in .NET How to show that $C=C[0,1]$ is a Banach space Let $C=C[0,1]$ be the space of all continuous functions on $[0,1]$. Define $\|f \|=\max \ |f(x)|$. I want to show that $C$ is a Banach space.     Below is my attempt and I was wondering if it's ok.    I know I have to show that $C$ is a complete normed space. Clearly, $\|f\| \geqslant 0$ and $\|f\|=0 \Leftrightarrow f=0$. $\|cf \|=\max~|cf(x)|=|c|\max |f(x)|=|c| \cdot \|f\|$. $\|f+g\|=\max~|f(x)+g(x)|\leq \max~|f(x)|+\max~|g(x)|=\|f\|+ \|g\|$. So $C$ is a normed space.   Next, I show that every Cauchy sequence in $C$ is convergent. Let $\{f_n\}$ be a Cauchy sequence in $C$. Let $\varepsilon \gt 0.$ Then $\exists$ an $N_1$ such that $$ \max~|f_n(x)-f_m(x)| \lt \frac{\varepsilon}{2}$$ for $n, m \gt N_1$ and $x\in[0,1]$.  But there is a subsequence $f_{k_n} $, which converges to $f$. So $\exists$ an $N_2$ such that $$ \max~\left|f_{k_n} - f\right|\lt \frac{\varepsilon}{2}$$ for each $n\gt N_2$.  Now Let $N = \max\{N_1, N_2\}$, if $n \gt N$ then $k_n \geqslant n\gt N$. So we have $$ \max~\left|f_n(x) - f(x)\right| \leqslant \max~\left|f_n - f_{k_n}\right| + \max~\left| f_{k_n} - f\right| \lt\frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon.$$  Thus, $\|f_n-f\| \to 0$ as $n\to \infty$. $\quad \square$  Thanks. My Command is below and its print Free CPU and Used CPU below format.  I want to restrict awk output to 2 decimal place.  grep 'cpu ' /proc/stat | awk '{usage=($2+$4)*100/($2+$4+$5)} {free=($5)*100/($2+$3+$4+$5)} END {print "Used Cpu:",usage "%"} {print "Free Cpu:",free "%"}'   Output :   Free Cpu: 98.9588%  Used Cpu: 1.04125%   Desired output :  Free Cpu: 98.95%  Used Cpu: 1.04% I don't like Hell, can I go somewhere else? I just opened a nether portal for the first time, and I must say, it's not a very good location, a sort of narrow trench/cavern thingy without any obvious clues as to which direction is "out".  I'm not willing to simply move my overworld portal (due to dilation it is unlikely to have much of an effect, anyway), and was wondering if there was a way to force minecraft to generate a new personal Hell for me to explore (hopefully one that doesn't involve me stuck in some wall!). Below, the  of second-order logic is used.    My question is about a second-order analogue of $ZFC$ other than the usual "second-order $ZFC$." Rather than define the latter, I'll just state that its (set-sized) models are exactly the $V_\kappa$s with $\kappa$ strongly inaccessible (or if you prefer, the uncountable Grothendieck universes).  Specifically, I'm interested in the second-order theory consisting of the usual (first-order) axioms of Pairing, Extensionality, Union, Choice, Infinity, and Powerset, and the Separation and Replacement schemes ranging over second-order formulas. I'll call this "$ZFC_2^{scheme}$."    Specifically, for each formula $\varphi(x, y_1,...,y_k)$ of second-order logic with only the displayed variables, each of which is first-order, we have a corresponding Separation and Replacement instance. We do not allow second-order parameters here, so formulas with second-order free variables don't yield Separation/Replacement instances.   The passage from $ZFC$ to $ZFC_2^{scheme}$ has a nice uniformity: it's an example of a more general operation $\mathcal{ZFC}$ which spits out a $ZFC$-analogue given a logic at least as strong as first-order logic.    This is the reason for the parameter restriction above: I want $ZFC_2^{scheme}$ to look just like $\mathcal{ZFC}(\mathcal{L})$ for any other logic $\mathcal{L}$, and not take into account the particular feature that we now have a new kind of variable at our disposal - and in particular, take as "syntax-free" an approach to logics as we can get away with. (This is a choice of arguable value, but it's the one I'm making for now.)   It's easy to show that every model of $ZFC_2^{scheme}$ is well-founded, so to understand its models we only need to look at transitive sets. A natural hope at this point is that $ZFC_2^{scheme}$ is just second-order $ZFC$ in disguise, that is, that for a transitive set $M$ we have $M\models ZFC_2^{scheme}$ iff $M=V_\kappa$ for some strongly inaccessible $\kappa$. However, this isn't at all obvious to me (although it is easy to show the right-to-left direction).     Question. What are the (set-sized) models of $ZFC_2^{scheme}$?   (Note by contrast that the "arithmetic analogues" $PA_2^{scheme}$ and second-order $PA$ are equivalent since each pins down $\mathbb{N}$ up to isomorphism - although that leads to .)  (I'm happy to drop Choice if that would help.)  EDIT: As Hanul Jeon pointed out below,  shows that consistently $ZFC_2^{scheme}$ (there called "$ZFC_2^{def}$") has countable models, so we have a partial answer. However, I don't see at the moment a way to get an outright $ZFC$ proof that $ZFC_2^{scheme}$ is strictly weaker than second-order $ZFC$. I'm connecting to some enterprise wireless network (eduroam).  The thing is that it saves the User/Password in the wireless settings, anyone can click 'Show password' and see my university account password.  What I want is that it prompts me everytime the credentials inputs when I try to connect to the network and don't save it under the 'settings' page.  Is this possible? If yes, how?  Thanks  Ubuntu 18.04 LTS What is the best way to share/distribute Excel macro? Is the sorting algorithm used by .NET's Array.Sort() method a  algorithm? Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups? In  (2017), Logan seems to be dying from sort sort of ailment, that prevents him to heal properly and seems to be decaying his body in a way that wasn't happening in any of the other films his character appears in.  I don't recall that this is being explained in the film. The Wikipedia page only mentions "adamantium poisoning" and "his healing factor fading", but it is not backed up with citations, so I would love to get an in-depth answer for this question, maybe using excerpts from the comic or original source material.  What is happening to Logan? How do I check if an element is hidden in jQuery? Is it possible to toggle the visibility of an element, using the functions .hide(), .show() or .toggle()?  How would you test if an element is visible or hidden? Number of surjective functions from A to B macOS Sierra restarting download instead of resuming Difference between a space and a "wave" symbol (tilde/"~")?