Finite dimensional field extension, finitely many intermediate fields
Intermediate fields between $\mathbb{Z}_2 (\sqrt{x},\sqrt{y})$ and $\mathbb{Z}_2 (x,y)$
Can some metadata about deleted posts be included in Data.SE? What does "xmlns" in XML mean? Delete votes from users who later become moderators shouldn't prevent community undeletion Rest mass of the photon Why is the rest mass of the photon is taken to be zero?  If rest mass of the photon is non zero, then it leads to infinite energy of the photon (by $E=mc^2$ equation) and there is a violation of conservation of energy. Is this a reason?  Also, is there any thing called the relativistic mass of the photon? How did the Death Star move? In the Star Wars saga, one of the most iconic weapons of the Empire was the Death Star: a self sustained weapons platform the size of a moon, with the capacity to destroy a planet. in A New Hope, we see the first Death Star in close proximity to the planet Alderaan, holding the planet hostage as the Empire tries to deduce the location of the Rebel Alliance's headquarters. In Return of the Jedi, we see the second Death Star in "dry dock" over the forest moon of Endor as it is being constructed.  My question is: How did the Death Star move? I imagine a spherical space station the size of a moon would be utterly massive, requiring a massive amount of thrust to move it. If it was constructed in low orbit of an inhabited planet/moon, the gravitational field would be immense, consisting of the planet/moon's gravity, and the Death Star's own considerable gravity. Breaking that gravitational field would be a massive task, yet we see no thrusters or engines on either Death Star. Were they hidden, stored under the station's surface while not in use? Was the station towed by another vessel? (a) $S_k=\{A\subset\mathbb{N}: |A|=k\}$ for $k\in\mathbb{N}$  (b) $S = \bigcup_{k=1}^\infty S_k$   Work: For (a), I am not too sure about what approach I should use. I think finding a bijective function between the $S_k$ and $\mathbb{N}$ is hard to do . I also thought about using the Schroeder-Bernstein theorem and find injective functions $f:\mathbb{N}\rightarrow S_k$and $g:S_k\rightarrow\mathbb{N}$ in order to prove that $S_k$ and $\mathbb{N}$ are numerically equivalent. However, I have a hard time finding and proving these injective functions. I haven't really looked at (b) as I have yet to complete (a). How do I add a Pokestop to an area that has none? It is an area where there are tons of businesses but unless I drive about a mile from my work area there are no Pokestops.   I want to add one since we have 1000's of people in the work area, a lot of them play Pokemon Go, and we all can't leave work to drive a mile away to get to a Pokestop area.   Plus it would be nice to be able to capture Pokemon on our breaks. I read this set of stories in the 80's.  What I recall is this:  A planet where the ruling caste was made up of humanoid women.  These women would, as almost pubescent girls, go out into the wilderness with only a spear and attempt to kill an animal in a life or death struggle.  When it happened, they would burst open a DNA packet in themselves that caused them to rapidly mature, their skin to become bronze colored, and give them some kinds of other "powers", I think over fire.  They were called "Bahronas", I think.  The population of this planet was primitive, but they were in contact with an advanced humanoid species, I think of actual homo-sapiens, who were there observing and advising them.  The advanced species wasn't allowed to intervene, just advise and watch.  The main character was a castaway boy, crash landed on their planet and stranded.  He became friendly with the ruling family of Bahronas and entered their confidence.  We find out through the course of the story that he's been genetically engineered and is not a normal boy.  At some point, the advanced species notices him and explains that he is a clone of a famous explorer and a spy, and they explain that they won't do anything in front of him.  The central part of the story is something of a blur, but, the final part of the story is told in a dramatically different style and shifts to a planet populated by catlike creatures who live in boughs of silk scarves, some of which sing.  THey are recording devices which play back the songs.  On this planet, there is another silk scarf, a white one, which doesn't play a song at all.  The cat creatures are greatly disturbed by it when they hear it's message, which they don't understand.  The narrator then goes on to say that it's a spoken word recording, made by the explorer which the boy was a clone of, explaining that he is captured and held hostage on the planet which makes the scarves, and forced to help make them (I think.) Names of some months don't make sense I'm not a native English speaker but I'm always trying to do my best. Unfortunately I have a real problem with dates for some odd reason, I couldn't learn when was my birthday until I was 12 years old.  Anyway, as I was learning names of the months on English, and since I know some Latin from high school I noticed that English months don't match up with their Latin numeral counterparts. For example:  September - 9. month October - 10. month November - 11. month December - 12. month  While on latin septem, octo, novem and decem are words for 7, 8, 9 and 10.  So, does anyone know why is that. At first I thought it had something to do with switch from Julian to Gregorian calendar but I figured it's too small of a difference.  Anyone? :) If $(A-\lambda{I})$ is $\lambda$-equivalent to $(B-\lambda{I})$ then $A$ is similar to $B$ I am a EE grad. student who has had one undergraduate course in real analysis (which was pretty much the only pure math course that I have ever done). I would like to do a self study of some basic functional analysis so that I can be better prepared to take a graduate course in that material in my university. I plan to do that next fall so I do have some time to work through a book fully. Could some one recommend some good books to start working on this?   Thanks in advance Regular Expressions: Is there an AND operator? Find a recurrence relation for the number of $n$-digit binary sequences with no pair of consecutive $1$s Find a recurrence relation for the number of $n$-digit binary sequences with no pair of consecutive $1$s.  I know my base case:$$a_1 = 2$$ for either $1$ or $0$. Normally to construct the recurrent relation I would say fill in $n-1$ spaces then add a $1$ or $0$, which is simply:$$a_n = 2\cdot a_{n-1}$$However, I don't believe this is right, because this doesn't take into account that there can't be any consecutive $1$s.  Even when I use the base case of $2$ digits:$$a_2 = 3$$for either $01$, $10$, or $00$, I still get the recurrence relation of:$$a_n = 3\cdot a_{n-2}$$which also does not account for $01$ and $10$ occurring consecutively.   So how would I express this? How to NOT show keyboard layout chooser popup when changing language in GNOME 3 Since I installed 17.10, when switching language with the assigned keyboard shortcut I get this switcher "window" that obstructs me from typing for a couple of seconds. And yes, a couple of seconds is a huge amount of time when switching languages a lot.      This happens both in X11 and Wayland but doesn't happen (the obstructive window will not appear) when I switch language by clicking on the lang indicator on top bar. Is there any way to tell this window "please go away"? Would it be possible for some to bend two elements? After reading this question  While it seems that it would break one of the rules of the show, that only the avatar can master all four elements. There would also be good reasons as to why someone couldn't learn all four, there is always the opposing element that is hardest for the Avatar to learn. I don't see any reason why someone couldn't learn to bend two elements (not master, but just bend since Thadeus's Answer  to the previous question is apparently the Definitive Answer on the subject.)    Many benders do learn to bend beyond their element such as Toph becoming a metal bender and Katarra bending blood. Now metal is considered part of the Earth but so is sand and it seems that sand is moved more like water. Iroh was able to adapt a water bender technique to fire bending which means it is possible to learn and use different bending styles. Also Iroh spent time connecting to the spirit world and that seems to be the main reason the Avatar is able to bend all 4 elements. So if someone was connected to the spirit world or even their past life that would fulfill that requirement.   Would it be possible to bend two elements?  There do seem to be times where it has been done in the series such as bending mud which is both Earth and Water. And there would definitely be limits such as a Earth bender couldn't bend Air and a Water bender couldn't bend Fire. But why couldn't a Fire Bender learn to bend Air. They both use breath control as a main focus of their bending. It also states on the Energybending wiki entry that all bending used to be done as an inner bending but that it became element bending.   So with new bending techniques being formed all the time, already stated used of combined elements being used, an assumed spiritual connection to a past life or a spiritual journey been taken, energybending rediscovered is it possible that two elements could be bent by one bender who is not the Avatar? How do I tell if matrices are similar? evaluate the limit as $x$ approaches $0$ of $(\cos x-1)/x$ without l'Hopital's rule I know with l'Hopital's rule it becomes $-\sin(x)$ which has the limit $0$. However, I have been wondering how to evaluate this limit without l'Hopital's rule. Simplest Example of a Poset that is not a Lattice A partially ordered set $(X, \leq)$ is called a lattice if for every pair of elements $x,y \in X$ both the infimum and suprememum of the set $\{x,y\}$ exists. I'm trying to get an intuition for how a partially ordered set can fail to be a lattice. In $\mathbb{R}$, for example, once two elements are selected the completeness of the real numbers guarantees the existence of both the infimum and supremum. Now, if we restrict our attention to a nondegenerate interval $(a,b)$ it is clear that no two points in $(a,b)$ have either a suprememum or infimum in $(a,b)$.   Is this the right way to think of a poset that is not a lattice? Is there perhaps a more fundamental example that would yield further clarity? I want create my own IDE but I want a code editor that would syntax highlighting.  I want to embed it as a control into a WPF window.  Malcolm Show that the boundary of a set equals the boundary of its complement $\newcommand{\bdy}{\operatorname{bdy}}$  I'm trying to show that $\bdy(A) = \bdy(A^c)$.  I know that $\bdy(A) = \operatorname{closure} A \setminus \operatorname{int}(A) = (\operatorname{int}(A^c))^c \setminus \operatorname{int}(A)$, but I don't know where to go from there.  Any help or hints would be very much appreciated.