let $K/k$ be a field extension and $a\in K$ algebraic over k. Prove that the subalgebra $k[a]$ is a field and that $k[a] = k(a)$ where $k(a)$ is the monogenic extension, i.e. the smallest subfield of $K$ containing $k$ and $a$.  I know that $a$ is algebraic on $k[a]$ since $k\subset k[a]$. I don't see how to go from there. Thank you.
If $\alpha$ is an algebraic element and $L \subset K$ are both field, does the polynomial ring $L[\alpha]$ is also a field?  I am trying to prove that the ring of fraction $L(\alpha)$ is equal to $L[\alpha]$. To do this, I use an exercise done in class : Let $K$ a field, $\alpha \in K$ and $L$ a subfield of $K$. Then $L(\alpha)$ is the smallest subfield of $K$ containing $L$ and $\alpha$.  Is anyone could help me?
Cube of an integer $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=k$ and $x, y, z, k$ are integers.  Prove that $xyz$ is cube of some integer number.    I was wondering about giving a parametrization for the rational points on the elliptic curve in $\mathbb{P}^2(\mathbb{R})$ $$ x^2 y + y^2 z + z^2 x = k xyz$$ but I am not too confident in the topic, so I am asking you to shed some light.  Thank you for your help. The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$. I try to show that the norm on the quotient space $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$, where $x = (x_n)_{n \in \mathbb{N}} \in \ell^{\infty} (\mathbb{N})$. My attempt is the following:     By definition, the quotient norm $\| \cdot \|_{\ell^{\infty} / c_0}$ is defined as   $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} = d( x + c_0 (\mathbb{N}), 0 + c_0 (\mathbb{N}) ), $$   which can be re-written as    $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} = \inf \{ \| x + y \|_{\infty} \; | \; y \in c_0 (\mathbb{N}) \}. $$   Since $0 = (0,0,...) \in c_0 (\mathbb{N})$, it follows that   $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} \leq \| x \|_{\infty} = \sup_{n \in \mathbb{N}} |x_n|. $$   I thought about using Bolzano-Weiserstrass' Theorem at a certain point, but I am not sure how to obtain the wished equality from here.    Any help is appreciated. Thanks in advance. No module named numpy I am trying to run Jupyter notebook from my Ubuntu subsystem in Windows 10. I've installed numpy, scipy, and pandas using the following commands:  pip install scipy numpy pip install pandas   When I load python in Ubuntu, there is no issue importing numpy in Ubuntu cmd.   But, when I run my Jupyter Notebook from Ubuntu terminal, and try to import numpy as np, or import pandas as pd, they report  ModuleNotFoundError                       Traceback (most recent call last) &lt;ipython-input-1-a9340201ed9f&gt; in &lt;module&gt;       5 import dash_html_components as html       6 import plotly.graph_objs as go ----&gt; 7 import numpy as np       8 from dash.dependencies import Input, Output       9   ModuleNotFoundError: No module named 'numpy'   Is there anyway to solve this issue? Thanks a lot. Is "do something by date X" inclusive? Possible Duplicate:          For example, if John says:      Return it to me by March 24th.    Does it mean that I need to return it to him before 23:59 p.m. March 24th, or before 23:59&nbsp;p.m. March&nbsp;23rd? How do I go about tracking down IO usage in Linux and other *nix OS'? Occasionally, something on my system starts hammering the disk. I would like to log data that would let me know about I/O usage per process, then I could go back to it later and see what keeps doing this. How should I go about solving this problem? Bonus points for solving this for arbitrary block devices. I want to run the command   python quast.py -o quast contigs.fasta   but I want to do this for multiple contigs.fasta files in multiple directories.  I'm running OSX. How do I integrate the Poisson equation to determine the electric potential along a particular direction (e.g., $z$)? This question is a sequel of sorts to my  about .  In the paper, the authors performed molecular dynamics (MD) simulations of parallel-plate supercapacitors, in which liquid resides between the parallel-plate electrodes.  The system has a "slab" geometry, so the authors are only interested in variations of the liquid structure along the $z$ direction.  In my previous question, I asked about how particle number density is computed.  In this question, I would like to ask about how the electric potential is computed, given the charge density distribution.  Recall that in CGS (Gaussian) units, the Poisson equation is   $$\nabla^2 \Phi = -4\pi \rho$$  where $\Phi$ is the electric potential and $\rho$ is the charge density.  So the charge density $\rho$ is proportional to the Laplacian of the potential.  Now suppose I want to find the potential $\Phi(z)$ along $z$, by integrating the Poisson equation.  How can I do this?  In the paper, on page 254, the authors write down the average charge density $\bar{\rho}_{\alpha}(z)$ at $z$:  $$\bar{\rho}_{\alpha}(z) = A_0^{-1} \int_{-x_0}^{x_0} \int_{-y_0}^{y_0} dx^{\prime} \; dy^{\prime} \; \rho_{\alpha}(x^{\prime}, y^{\prime}, z)$$  where $\rho_{\alpha}(x, y, z)$ is the local charge density arising from the atomic charge distribution of ionic species $\alpha$, $\bar{\rho}_{\alpha}(z)$ is the average charge density at $z$ obtained by averaging $\rho_{\alpha}(x, y, z)$ over $x$ and $y$, and $\sum_{\alpha}$ denotes sum over ionic species.  The authors then integrate the Poisson equation to obtain $\Phi(z)$:  $$\Phi(z) = -4\pi \sum_{\alpha} \int_{-z_0}^z (z - z^{\prime}) \bar{\rho}_{\alpha}(z^{\prime}) \; dz^{\prime} \; \; \; \; \textbf{(eq. 2)}$$  My question is, how do I "integrate the Poisson equation" to obtain equation (2)?  How do I go from  $\nabla^2 \Phi = -4\pi \rho$ to equation (2)?  In paricular, where does the $(z - z^{\prime})$ factor come from?   Thanks for your time. How can I convert timestamps in a column to a date? I have a file containing this:  1415602803,LOGIN SUCCESS,AUTH,user2,192.168.203.63,10.146.124.73,59996,22 1415602807,LOGIN SUCCESS,AUTH,user1,172.24.31.10,172.32.1.1,48191,22 1415602811,LOGIN FAILED,AUTH,root,172.24.166.153,10.146.124.73,52506,22 1415602815,LOGIN FAILED,AUTH,user3,192.168.123.55,10.146.32.99,55750,22   I want to convert the timestamp to a date in this format:  2014-11-10 02:00:03,LOGIN SUCCESS,AUTH,user2,192.168.203.63,10.146.124.73,59996,22 2014-11-10 02:00:07,LOGIN SUCCESS,AUTH,user1,172.24.31.10,172.32.1.1,48191,22 2014-11-10 02:00:11,LOGIN FAILED,AUTH,root,172.24.166.153,10.146.124.73,52506,22 2014-11-10 02:00:15,LOGIN FAILED,AUTH,user3,192.168.123.55,10.146.32.99,55750,22   How can I do that?  I know this works: perl -pe 's/(\d+)/localtime($1)/e' (from ) but the output format is Mon Nov 10 02:00:03 2014.  I know this command can convert timestamps into my desired output: date -d@1415602803 +"%F %H:%M:%S", but I couldn't make it work with awk using system("cmd") because of all the quotations and whatnot. How to install PYPY3 on Ubuntu, for newbies? I want to install pypy3 in Ubuntu. I have read the answer for this question "How to install PyPy3 (2.1, beta) on Ubuntu?" and still have no idea what to do. Could some kind soul please explain it so that even I can understand it :)  Here's what I have done so far:  Went    Read that I had to go   Downloaded: pypy3-2.1-beta-linux_x86_64-portable.tar.bz2  Opened it with the Archive manager (because it seemed like a reasonable thing to do)  Extracted it to desktop/PYPY3  Then desperately tried all the shell commands I came across last night. I will not be able to give a clear account of what I tried and what errors came back, as it is kind of a blur to me at this point.  But I can tell you that I got the Tar thing unpacked at one point and that I have tried running the pypy executable from the command line from the folder containing it, but got this:   bash: /usr/bin/pypy: No such file or directory   Could someone please tell me what to do. (have read the readme, the install docs at pypy.org and lots of posts) Add an "I don't understand" option for review audits Can players on a PC and an Xbox play together in any way? I.e. can Xbox users connect to a (modfied?) PC server, or vice versa? I have always believed that it was ok to replace a capacitor with a slightly higher voltage rated one. eg 16v with 25v.Going higher than this would risk the capacitance value being out at the working voltage. However I just read the following quote from this       Electrolytic capacitors are non-ferroelectric with a very low dielectric constant. Their capacitance is derived from a very high surface area and nanometer thick dielectric layers. Their capacitance is not a function of applied voltage.‐Design Solutions For DC Bias In Multilayer Ceramic Capacitors (August 2010 Electronic Engineering Times)   So does this rule not apply to aluminum electrolytics? Eg will I have no issues replacing an electrolytic 25v 100uf with a 50v 100uf assuming both are low esr, similar footprint and good quailty? biblatex: remove commas between last and first names in bibliography Probably a really simple one this - I'm starting out with C# and need to add values to an array, for example:  int[] terms;  for(int runs = 0; runs &lt; 400; runs++) {     terms[] = runs; }   For those who have used PHP, here's what I'm trying to do in C#:  $arr = array(); for ($i = 0; $i &lt; 10; $i++) {     $arr[] = $i; } Citing two authors gets displayed incorrectly Moving an already-running process to Screen My office job routinely sees me connected to a Linux box via VNC.  Sometimes I start a remote job on the console, and realize later that it runs much longer than expected. (Should have started that one under Screen in the first place...)  I don't want to keep my workstation running overnight just to keep the VNC session open; I would like  to move that already-running remote job into a Screen session (on the remote box), so I can power down the workstation (and reconnect next morning).  How can this be done, if at all? An integer $a \pmod m$ has inverse if and only if $\gcd(a,m)=1$? An integer $a \pmod m$ has inverse if and only if $\gcd(a,m)=1$?  Why is this? I tried understanding it from my notes but I don't get it.  A thorough explanation would be greatly welcome.  Thanks for all the answers so far. I think my biggest problem is understanding why $\gcd(a,m)\neq1 \Longrightarrow \not\exists a^{-1}$. In .NET, which loop runs faster, 'for' or 'foreach'? Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice? Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice?  This question is answered here:   I'm not interested in a proof itself of this theorem, I want to know why this requires the Axiom of Choice. LWJGL—How to Animate and Render Characters Using a Sprite Sheet I'm making an RPG similar to Super Nintendo RPGs in the early '90s. I'm handling movement much like Squaresoft's Final Fantasy IV.   Notice that the player never actually changes his x and y positions on screen. What I'm asking, to be precise, is how could I, in LWJGL through user input (say, the arrow keys), take the sprite sheet I have of my character's movement animations and render them in a [partially] fluid animation like above?  I've been perusing the internet for hours, and I've not found a surefire way to do this. I don't have any code as of yet; I'm more or less looking for a guideline or some pointers in the right direction concerning how I might tackle this.  And, to note, this'll be a rather large game, so I want to focus partially on speed and efficiency—that meaning that I don't want to dive into Slick2D and libGDX unless absolutely necessary.