I have already been able to show that if the irreducible polynomial $ f $ has degree n, then its roots are $ a, a^q \ldots, a^{q^{n-1}}$, where $ a $ is a roots of $ f $ to some extent.  How can I show that they are all different?
Every irreducible polynomial $f$ over a perfect field $F$ is separable.   Can you check my proof? Assume that $f$ is irreducible but inseparable. So we have $f=\sum_i h_ix^i$ and $f^p=\sum_i h_i^px^{ip}$. Now I use Frobenius mapping $\phi (f)=f^p=(\sum_i h_ix^i)^p=\sum_i h_i^px^{ip}$ so it is reducible. Contradiction. Should I also show for char(F)=0?
how to prove $G_1$ and $G_2$ aren't isomorphic? Atomic Physics: stimulated emission I'm studying a chapter about atomic physics right now but there's thing I just don't seem to understand. When stimulated emission occurs, there's an incoming photon which stimulates the atom to go an energy level lower with the result that a new photon gets emitted.  Can someone explain the reason why the incoming photon stimulates the atom to emit a second one? To me it seemed more logic for the atom to absorb the photon and go an energy level up. There seems to be a phenomenon which I don't know about. I came to know about , it's a website that allow search for flights by country. I mean you just enter from like "Germany" and to like "Belgium" and you get best offers from all airports between the two countries. That is great. And I was able to arrange a good-price trip to Brussels.   Now I want to arrange another trip to Turkey, but Whichbudget is not helping much this time. And I can't find any other websites that allow "search by country".   Update: Not duplicate to  as his question is broad to the extent of a state. But my question is broad to the extent of a whole country + less price + "Any Time". If anyone checks , he'll understand what I'm saying. Especially that whichbudget allows "Any Time" option, not just +/-3 days.  Does anyone know more similar websites? In an ideal case, we assume strings and ropes to be massless and having same tension throughout.  However, I was doing some questions which involved a rope with mass.  The question was that if a uniform thick rope of length 5 m is resting on a horizontal smooth (assume frictionless) surface and is pulled by a force of 5N from one end, then what is the tension in the rope at the point 1 m away from the end where force is applied? (sorry for the poor image)    I came to the conclusion that since rope is not fixed, tension anywhere should be zero. However, when I checked the answer, it was 4 N. I failed to understand how.  Then the next part of the question was that if the rope now hangs between two fixed points which are at same level, then what would be the tension in the rope at its lowest point?    The answer has to be expressed in terms of mass of string m, acceleration due to gravity and $\theta$ where $\theta$ is the angle that tangents to the rope make with the horizontal.  I took that force acting on lowest point is mg, since it is center of mass and hence T = mg. However, according to the answer it depends upon $\theta$ as well. However, I failed to see what I was doing wrong?  I would appreciate any help in solving these questions, since I have no clue of where exactly I am wrong. I did search the Internet and a few books, but everywhere they assumed string/rope to be massless. How can I prove this sequence is decreasing? My problem is how to prove  $$\frac{1+2}{1}  &gt; \frac{1+3+\frac{3^2}{2!}}{1+2} &gt; \frac{1+4+\frac{4^2}{2!}+\frac{4^3}{3!}}{1+3+\frac{3^2}{2!}}&gt;\cdots $$  i am sure that the sequence is decreasing as limit exp(1).  However, the reason why the sequence is decreasing is not evident.  Thanks if you consider this issue and make me relief. I'm looking to create a simple short-lived reservation system, and I'd like to generate confirmation numbers that are   unique random-looking alphanumeric short-ish, at least much shorter than 32 character-long strings returned by sha1   I'm only looking to have ~500 reservations, so I don't imagine high likelyhood of collissions.  One idea I had is generate an sha1 hash based on a date-time stamp and username, then truncating it to its first 10 characters.  Would something like that be reliably unique enough for the purposes of processing ~500 reservations? I'm attempting to install Ubuntu 14.04.1 (the only version I have handy) onto my old Compaq Presario R3000 (this Ubuntu was recommended for this computer). The installation has stalled at about 60% and hasn't moved in an hour.  The message at the bottom says: Configuring bcmwl-kernal-source (i386).  I have no experience with Ubuntu (or with Linux, for that matter), coming from a strictly-Windows background). Can anyone offer some advice? If $2^n - 1$ is prime from some integer $n$, prove that n must also be prime. ASP.NET -> WCF concurrency problem Capture characters from standard input without waiting for enter to be pressed I can never remember how I do this because it comes up so infrequently for me. But in C or C++, what is the best way to read a character from standard input without waiting for a newline (press enter).  Also ideally it wouldn't echo the input character to the screen. I just want to capture keystrokes with out effecting the console screen. Do lens hoods need to be matched to focal length? Is it true that not all generic lens hoods work on all lenses (provided the filter diameter matches, of course)? I read that the focal length plays a role in it - but I cannot find such readings on any lens hoods I see available online. Listing layers from .mxd file using ArcPy for ArcGIS Pro? Strikeout in math mode How can I strikeout words in math mode (similar to \sout with \usepackage{ulem} in text mode)? I've tried \usepackage{cancel} and \(\cancel{1+1=2}\) but that draws a diagonal line. This is a spin-off question from .   says this on the subject of storage order (clustered index) and the output order     A lot of people believe that a clustered index guarantees a sort order on output. But that's not what it does; it guarantees a storage order on disk.   See, for example, .   I've read the blog post by Hugo Kornelis and understands that an index doesn't guarantee that the sql server reads the records in a specific order. Yet I have a hard time accepting that I can't assume this for my scenario?  CREATE TABLE [dbo].[SensorValues](   [DeviceId] [int] NOT NULL,   [SensorId] [int] NOT NULL,   [SensorValue] [int] NOT NULL,   [Date] [int] NOT NULL, CONSTRAINT [PK_SensorValues] PRIMARY KEY CLUSTERED  (   [DeviceId] ASC,   [SensorId] ASC,   [Date] DESC ) WITH (     FILLFACTOR=75,     DATA_COMPRESSION = PAGE,     PAD_INDEX = OFF,     STATISTICS_NORECOMPUTE = OFF,     SORT_IN_TEMPDB = OFF,     IGNORE_DUP_KEY = OFF,     ONLINE = OFF,     ALLOW_ROW_LOCKS = ON,     ALLOW_PAGE_LOCKS = ON)   ON [MyPartitioningScheme]([Date])   My original query was this:  SELECT TOP 1 SensorValue   FROM SensorValues   WHERE SensorId = 53     AND DeviceId = 3819     AND Date &lt; 1339225010   ORDER BY Date DESC   But I suggest that I could as well use this one (read below for my explanation):  SELECT TOP 1 SensorValue   FROM SensorValues   WHERE SensorId = 53     AND DeviceId = 3819     AND Date &lt; 1339225010   As you can see, my table rows are small (16bytes) and I've got only one index, a clustered. In my scenario, the table consists of 100.000.000 records at this moment (and this will most likely increase tenfold).  When the database server queries this table it has two ways of finding my rows, either it seeks the primary key and thereby reading and returning my values in desc. order of Date, or it has to do a full table scan. My conclusion is that a full table scan on all those records will be way too slow and the database server will therefore always seek the table via its primary key and thereby returning the values sorted by Date DESC Where has the trailing newline char gone from my command substitution? Mean and variance of inverse of a normal RV My random variable $X$ is normally distributed with mean $b$ and variance $p$. I defined a new RV, $Y$, such that $Y=1/X$. Does anyone know how to find the mean of $Y$ and $Y^2$? using variable length argument in scala Select from list of reasons when performing downvotes Preamble: I know that 'give a reason when downvoting' is a very common request, and I assure you that I have read these posts and the discussions in them and my request is very different. This isn't about encouraging people to comment and is more about gathering metrics.  I'd like to see a feature whereby people who are downvoted get some better feedback on why they were downvoted, especially given that people rarely leave comments in practice.  So my suggestion is that when user clicks on the downvote, they are prompted to pick from a short list (No more than 5) of common reasons. It would be a popup that replaces the existing "Please consider adding a comment" box.  The user receiving the downvote would see how many people downvoted them for each reason, but not who did it. It would significantly address the issue of people being too lazy to comment or not wanting to comment (for fear of retaliation or any other reason) while helping to ensure the user gets useful feedback.  Perhaps the downvote could also be weighted based on whether or not a reason is picked.  For questions the list might look something like:   Not a question Not relevant to the subject area Trivial question with no research effort Poorly asked question Other   And for answers, this might be:   Nonconstructive or not an answer Incorrect or dangerous answer Poor explanation Doesn't answer the original question Other Endpoint of a line knowing slope, start and distance In a Cartesian system, I've got the slope, start point and distance of a line segment.  What's the formula to find the endpoint? Using pgfplots, why do I get "undefined control sequence" when trying to use a foreach variable as a parameter to addplot? Since I have multiple plots with error bars, I would like to shift the error bars slightly to make the overall plot more readable. However, when I try to do this using the variable of a \foreach as the amount to shift, I get an "undefined control sequence" error.  \documentclass{article}  \usepackage{tikz} \usepackage{pgfplots}  \begin{document}  \begin{tikzpicture} \begin{axis} \foreach \x/\y in {a/-1cm, b/0cm, c/1cm} {     \addplot+ [         %every error bar/.append style={xshift=\y},         %every node/.style={xshift=\y},         error bars/y dir=both,         error bars/y explicit,     ] coordinates {         (0, 0) +- (0, 1)         (1, 0) +- (0, 1)         (2, 0) +- (0, 1)     }; } \end{axis} \end{tikzpicture}  \end{document}   (The document does not compile when I uncomment either one of the two comments.)  What's wrong here?  EDIT: Added another variable to the \foreach. In my actual use-case it would contain the filename to plot data from or the label to use for the curve.