Prove that $D_n$ is generated by ⟨r,f⟩
Show the Dihedral Group $D_n$ is generated by rotations and reflection along the x axis.
Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$ I don't think I fully understand.  Let's say there is a root $x_0 \in K=\mathbb{F}_p(t)$, where $p$ is a prime number.  Then $x_0 = \frac{P(t)}{Q(t)}$ for some polynomials $P,Q \in \mathbb{F}_p[t]$. We can assume $\gcd(P,Q)=1$  and $x_0^p-t= \frac{(P(t))^p}{(Q(t))^p}-t= \frac{P(t)^p-tQ(t)^p}{Q(t)^p} =0$, so coefficients of $P(t)^p, tQ(t)^p$ must be identical, which contradicts $\gcd(P,Q)=1$,   hence such $x_0$ does not exist and the polynomial has no root in $K$.   am I about right?    anyway, I would appreciate an explanation about $\mathbb{F}_p(t)$, what is $t$? what is the meaning of a variable which does not belong to any specific "world"? I cannot use $t$ as if it was a member of $\mathbb{F}_p$ and hence cannot assume $t^{p-1} = 1 \pmod p$... How to get git-completion.bash to work on Mac OS X? I have followed  to  add git-completion.bash to my /opt/local/etc/bash_completion.d/git-completion  and I put PS1='\h:\W$(__git_ps1 "(%s)") \u\$ ' in my .bashrc_profile  But now I am getting this -bash: __git_ps1: command not found everything I do a cd.  Can you please tell me what am I missing? resizable serial console window? How do the Chitauri "ships" propel themselves? In the 2012 The Avengers film, the Chitauri army arrives through the portal along with these big robot alien whale Chitauri dudes. Here's one chasing Iron Man:  These guys basically swim through the air like... giant robot alien whales. I don't see any means of propulsion. Other, more humanoid Chitauri soldiers fly around on things that seem to be mechanical &quot;speeders&quot; of sorts. What are these guys? Are they ships, creatures, or both? Is there an explanation anywhere in the Avengers universe for how they defy gravity and swim through the air? Why does C++ have header files and .cpp files? There are many beings that have very big powers. There are beings that are worshipped as gods (Asgardians; and I would lump Azazel into that as well since Marvel seems to imply that he's - while a mutant - someone people consider to be "Devil"). There are mutants whose power would clearly have been considered as "god like" by people worshipping Greek or other polytheistic pantheons where the threshold to what made you a god was pretty low.  But are there any actual major "Gods" in Marvel universe?  Since there are many different definition of what a "god" is, the one that is pertinent to this question:   Power wise, being able to create Universe(s); e.g. gods more in a sense of Christianity/Judaism scale rather than "Pan" or "Hestia" in Greek pantheon who merely have some limited supernatural powers over small domain of natural events. Effectively immortal, at least excluding specific exotic attacks. Origin-wise, be "supernatural", not merely a human mutant or a powerful alien with advanced technology. Desire/demand/like followers' worship or prayers. After purge mussels 6 time in salt water 20 mins each, why they still exude much grime? I  buy mussels at Pusateri and Whole Food Market for a year in Toronto. They sell them on ice and dry, not in running water.      , Bart teaches add salt to cold water and  soak for 20 mins. But I repeat 6 times...so 20 mins x 6 times = 120 mins of soaking!!! Mussels exudate less black grime every next time, but even after 6 times, water still has much black grime! See my pic under! Why? This happening for a year!     Part 2,  has GIF that shows you what I mean. how to change "§" (section symbol) key to write "`" (back quote) I'm using back quote many times but never use the '§' symbol.  I'd like to change the key to always write back quote instead. does anyone know how can I change it programtically? I'm trying to figure out the best way to model the sphere-spiral here:    Is it possible to make this shape directly? Or do you have to make it as a cylinder and wrap it to a sphere? Why is everything floating in space so cold? Of course there is pretty hot stuff too. For example Suns.  But isn't the vacuum of space the perfect containment for heat? And shouldn't the rare collision with particles even heat up an object that floats around in space?  Why aren't they hot from time to time? Is it because they loose their heat by radiant heat?  And if so does radiant heat emission always occur until the object is the same temperature as the surrounding or does it depend on other factors? Keep items in inventory on death in Minecraft I once played a vanilla Minecraft adventure map (called "Herobrine's Mansion") that prevented the player from losing their items on death. Which commands can be used to prevent the player from losing their items on death (in vanilla Minecraft)? I'm trying to find a good way to print leading 0, such as 01001 for a . While the number would be stored as 1001, what is a good way to do it? I thought of using either case statements or if to figure out how many digits the number is and then convert it to an char array with extra 0's for printing, but I can't help but think there may be a way to do this with the printf format syntax that is eluding me. Polynomial in $n$ variables Does tourist visa (90 days) apply after a long-term visa ends in Schengen countries? Are black holes in a sense transparent? Meaning, they don't 'hide' any stars that happen to be 'behind' them?  A regular object obviously blocks a point source of light emanating from 'behind' it. Meaning it creates an area of space photons cannot get to us from. But a massive object curves space, and so via gravitational lensing we can sometimes see distant stars that are 'behind' other stars (by behind I mean that if the star in the middle would suddenly vanish, the trajectory of photons traveling from the distant star to Earth would have passed through the volume now occupied by the middle star).  My thinking, is that black holes curve space so much, that from ANY source of light outside it, light has a trajectory to reach us on Earth (assuming of course no other object blocks it on its way).  The reasoning for this is: Suppose we draw the straight line between us and the light source (as if the black hole wasn't there at all). Photons from our light source crossing the event horizon would obviously 'fall' inside. A photon in theory traveling exactly tangent to the event horizon would in a sense go 'into orbit' around the black hole (obviously there is no such thing, because the black hole isn't perfectly spherical, and there are quantum effects and so forth. But I'm focusing on the geometry), crossing our straight line exactly at the intersection of our line and the event horizon. Any photon passing near the event horizon, but a bit farther, would follow a curve that intersects our line a little farther from the event horizon. Continuing this, there would be at last be a photon that would exactly curve our way. Using Gimp to batch convert images to another format in Windows As implied by the title, does anybody know how to use the gimp-console-[version].exe program to batch convert images between formats (with default settings) in Windows ? build folder usage issues What can cause an answer to get deleted without information about who actually deleted it? Usually when answer gets deleted directly (i.e. not as side effect of its parent question deleted) it says "deleted by [name here] [time stamp]". Plus, it always leave a revision.  However, I just stumbled over  on Stack Overflow:        Weird, don't you think? Looking in the , nothing strange in there and nothing strange in  as well.  Finally, the last straw... :        Another thought was maybe some automagical conversion to comment, but no such comment exists.  Can anyone shed some light over this phenomenon? Is it legal to add your publications to ResearchGate? I recently created a profile on . And to add your publications list it requires you to upload the paper. Is it legal to upload published journal articles on public servers, where everyone can download it for free? I don't need a recommendation or a list but my knowledge of GPS and "navi"s and the differences between them is an utter mystery to me.  But often I'm lost in backstreets in random countries where access to something like a live Google Maps would be awesome. I absolutely cannot afford a smartphone with worldwide roaming though.  Can GPS's do what I want or am I barking up the wrong tree? If they can is it only high end ones than can?