I am trying to find an irreducible polynomial of degree 3 in Z5 and then use that polynomial to construct a finite field. I found that the polynomial $x^3 + x^2 + 1$ is irreducible in Z5 (since none of 0, 1, 2, 3,4 are roots). But I'm stuck on how to use that polynomial to construct a finite field.
I want to construct a field with $125$ elements. My idea is to consider the polynomial ring $\Bbb F_5[x]$. It is enough to find an irreducible polynomial $f\in \Bbb F_5[x]$ of degree $3$ because then $\Bbb F_5[x]/(f)$ is a field with exactly $5^3=125$ elements.  How do I find an irreducible polynomial of degree $3$ in $\Bbb F_5[x]$?
problems installing mathtools Was Tracy Mills a part of John Doe's plan from the beginning? In , John Doe is shown as being extremely meticulous and patient while carrying out his plan of murdering and/or torturing people in accordance with the seven deadly sins.  At the end of the film, it is revealed that John Doe had killed and decapitated Tracy Mills, the wife of Detective David Mills.  Was Tracy Mills his intended target from the beginning?  Did John Doe decide to kill Tracy only after Detective Mills and Somerset discovered his residence? Does every Abelian group admit a ring structure? How to execute a program or call a system command from Python How do you call an external command (as if I'd typed it at the Unix shell or Windows command prompt) from within a Python script? Importance of normal distribution Why did the normal distribution become such a popular (important) distribution? I know one reason is because of CLT. Can you please give more reasons? After monitor calibration colors on my two monitors are still different, why? I have a pair of pretty nice monitors (HP ZR30w) which I calibrated with a pretty good calibrator (i1-Pro) and after I calibrated both they still show colors every so slightly differently.  My left monitor has a little more magenta color to it in photos.  Is monitor calibration only ment to get things 'close'?  Other people in the office have experienced similar issues with their pairs of monitors, both HP ZR30w and Dell 3007WFP-HC monitors.  If I can't even get colors to match between my two exact same monitors connected to the same computer I have little confidence that colors on my monitors will look the same on somebody else's calibrated monitor :p What is the easiest/best/most correct way to iterate through the characters of a string in Java? StringTokenizer? Convert the String to a char[] and iterate over that? Something else? Complexity of $T(n) = 2T(n/2) + n$ How to detect existence of a class using SFINAE? grep strange behaviour with single letter words I am removing stop words from a text, roughly using this   I have the following  $ cat file file types extensions  $ cat stopwords i file types   grep -vwFf stopwords file  I am expecting the result: extensions  but I get the ( I think incorrect)  file extensions   It is as if the word file has been skipped in the stopwords file. Now here's the cool bit: if I modify the stopwords file, by changing the single word/letter i on the first line, to any other ascii letter apart from f, i, l, e, then the same grep command gives me a different and correct result of extensions.  What is going on here and how do I fix it?  I'm using grep (BSD grep) 2.5.1-FreeBSD on a Mac OSX GNU bash, version 4.4.12(1) Show question ID When I want to close a question as duplicate using the app, I have to enter the question ID as usual. Problem is, as far as I can tell, the question IDs are shown nowhere! Please add the ID somewhere. SQL query to convert columns into rows how to create a roster sheet in sharepoint 2010 Is it safe to format msftres, msftdata and hidden partitions? Cannot boot because: Kernel panic - not syncing: Attempted to kill init! This is what I've got. I cannot boot and I see the two of three little lights on the keyboard blinking all the time and the cooling is working on full power, almost.  I saw in other askubuntu topic suggestion for memory test. I did it. I didn't have any errors.  I am 99,9% sure it is not a hardware problem, because I have Windows 7 and I can boot it and it works just fine.  This problem came without any warning or something. I just turned off the laptop one day and at the other day I couldn't boot.   From recovery mode I get this: How to interpret these acf and pacf plots Following are acf and pacf plots of a monthly data series. The second plot is acf with ci.type='ma':    The persistence of high values in acf plot probably represent a long term positive trend. The question is if this represent seasonal variation?  I tried to see different sites on this topic but I am not sure if these plots show seasonality.           Edit: following is the graph for lag upto 60:     Following are plots of diff(my_series):     And upto lag 60:     Edit: This data is from:  Here the contributors did not consider acf and pacf plot of original or differenced series worth mentioning (so it must not be important). Only acf/pacf plots of residuals was referred to in a couple of places. Is it safe to answer "erase disk and install Ubuntu" on a virtual machine? Update 2014-04-17 04:47 UTC: Stable, with some odds and ends to clean up      The main MSO/MSE sites are stable, but there are plenty of corners left to sweep. For details, see:    Not all of them mind you, just Meta Stack Overflow into Meta Stack Exchange.  I'll have you know, factually, that we really did believe this would take place in six-to-eight weeks when it was , but we've finally got a tentative date established.  If things go according to plan, we'll begin the operation shortly after 8:30 PM EDT on the 16th, completing everything within a few hour window. Ideally, all service disrupting endeavors will be completed within the first hour, which include:   The availability of Meta Stack Overflow The availability of Stack Overflow (more on this shortly) The availability of Chat on Meta Stack Overflow   As you know, we'll be bolting on a standard issue per-site meta on Stack Overflow, while breaking the link between what will have become Meta Stack Exchange and Stack Overflow. There's some irritable innards in that particular region, so we'll be asking folks to expect the unexpected during the maintenance window. Affected sites will be in and out of read-only mode, or quite possibly just down for maintenance while we're doing the actual break.  I'll be posting a system message on both Meta Stack Overflow as well as Stack Overflow itself 24 hours before we begin the actual work.   Did I say tentative in a six-to-eight-something time frame again? Why yes, I did - we're currently running some tests on the dev tier to simulate the split as closely as possible, so that we can better anticipate quirks that we need to address prior to actually doing it.  Once the sites have actually been stood up functionally on the production tier, we're going to get to work on finishing touches. This will include things like site settings (which determine bits of text you see in the question prompts, rep thresholds, etc) and all sorts of other fine tuning.  Once we're done:   Stack Overflow will have a blank, empty child meta site just like every other site The new Meta Stack Exchange will be up and running All rooms on Chat Meta Stack Overflow will be moved to Chat Meta Stack Exchange  I'll send over about 150 - 200 questions from MSE to MSO, most of which will be tag or SO-specific feature related posts that got buried by the network stuff without resolution.   If the tentative date moves, I'll update this post with the new date. However, we do not anticipate this happening any later than Saturday, April 19 - in the case that we see down time we can't mitigate and need to do it on a weekend.  The ground work is basically done, we're finishing up sanity now with a great deal of confidence, and we're pretty certain that we'll hit between those dates, if not directly on the tentative 16th.  Questions? Kvetch? Anonymous confessions? Answer below, and I'll address each in comments. Cannot vote to close immediately after edit Why did sampling boost the performance of my model? I have an imbalanced dataset with 88 positive samples and 128575 negative samples. I was reluctant to over/undersample the data since it's a biological dataset and I didn't want to introduce synthetic data. I built a Random Forest Classifier with this original dataset. I got an F1 score of 0 for the positive class. Zero precision. Zero recall. I cross-checked the predictions and test data. The model predicts some positives none of which are actually positive. Worst performance.  So, I tried to oversample the positive class. I upsampled the positives to 1000 samples. To my surprise, the F1 score for this dataset was 0.97, for the positive class. Then I tried lesser samples. I was able to achieve an F1 score of 0.83 with 200 positive samples, which is just 2.25 times of the original positive samples.  I would like to know why this occurs. For 88 samples, F1 score is 0.00 (rounded off to two digits). For 200 samples it's 0.83. There is no data leakage. All the features are engineered. I used imbalanced-learn module for oversampling. Can someone explain why is this difference in performance?