Let $E|F$ be an algebraic field extension and a ring $K$ such that $F\subseteq K\subseteq E$. It is true that $K$ is a field?
This is an exercise in some textbooks.     Let $E$ be an algebraic extension of $F$. Suppose $R$ is ring that contains $F$ and is contained in $E$. Prove that $R$ is a field.   The trouble is really with the inverse of $r$, where $r\in R$. How to prove that $r^{-1}\in R$, in apparent lack of a characterization of $R$.  It occurred to me to use the smallest field containing $R$ ($R$ is easily shown to be an integral domain), that's the field of quotients, and proving that it's $R$ itself, but I don't really know how to proceed.  A not-too-weak, not-too-strong hint will be much appreciated.   Beware $ $ Readers seeking only hints should  beware that there is now a complete answer.
Solving $x - a \log(x)=b$ Let $a&gt;0$ and $b \in \mathbb{R}$: Assume there exists an $x &gt;0 $ s.t. $$x - a\log(x) = b$$ holds. How can it be determined in closed-form? When i tried to install PIP for python3 in Ubuntu 18.04.2 LTS  Command I used:   root@DESKTOP-0BEE3OD:/mnt/c# apt install python3-pip  E: Could not read response to hello message from hook [ ! -f /usr/bin/snap ] || /usr/bin/snap advise-snap --from-apt 2&gt;/dev/null || true: Success   I am not able to fix this error. How to influence the name of the pdf file created with pdfLaTeX (from within the source code)? I have a source file called 2011-05-27_Myfilename.tex and I'd like to create a PDF file from it called only Myfilename.pdf.  Can this be configured from within the .tex file itself?  (It seems to be possible with    pdflatex -jobname=Myfilename.tex 2011-05-27_Myfilename.tex   However, as many different files are concerned, It would be easier to have an option like output=Myfilename in the LaTeX source code.) Distribution of the ratio of two i.i.d standard normals I am reading an article in which the author states that given two i.i.d random variables $X,Y\sim\mathcal{N}(0,1)$, we have $\mathbb{P}(\frac{X}{Y}\le t)=\mathbb{P}(\frac{X}{|Y|}\le t)$ since the random varaibles $\frac{X}{Y}$ and $\frac{X}{|Y|}$ are identically distributed by the symmetry of the standard normal distribution.  But I don't really understand why the random variables $\frac{X}{Y}$ and $\frac{X}{|Y|}$ are identically distributed. Can you provide a reason for why these two random variables are identically distributed? Update on Ubuntu - coloured lines around everything Everyday I check for updates on Ubuntu 16.04 LTS. I updated to 4.4.0-65-generic today and now on everything I'm getting thick rainbow coloured lines around every dialog box in every program. My setup is pretty standard. Skylake i5-6500, Nvidia 950gtx. Does anyone know how to fix it? Let $\alpha$ be an irrational real number and let $a_j$ be a sequence of rational numbers converging to $\alpha$. Suppose that each $a_j$ is a fraction expressed in lowest terms: $a_j = \alpha_j / \beta_j$. Prove that the $\beta_j$ tend to $\infty$  Attempt: AFSOC that $\beta_j$ does not tend to infinity, then it is bounded by some $M$. We can also find an interval $\alpha \in (k, k+1)$. Let $\epsilon &gt; 0$, $\exists N_0$ such that for all $N &gt; N_0$ we have $|a_N - \alpha|  &lt; \epsilon$. So $|\frac{p_N}{q_N} - \alpha| &lt; \epsilon$. We try to use the fact that $q$ is bounded but fail to derive a contradiction. I do not know how to use the fact that $\alpha$ is between two integers, although it might not be relevant at all.  Ideas? Could anyone give me an explanation of the meaning of "get one's finger in the air" and the usage of it? In the review page, it shows stats of recent reviewers. But it is showing wrong stats. Because today (on Monday) it shows week count 53 (on Meta Stack Overflow) which is impossible as there is a limit of 20 reviews per day.    PS: This bug   and . But the bug has occurred again. Why double-headed arrows when the direction of node is specified? I am quite puzzled when drawing the connection of nodes.  \usepackage{xcolor} \definecolor{darkblue}{HTML}{1F4E79} \definecolor{salmon}{HTML}{FF9C6B} \tikzstyle{cell} = [rectangle, minimum width=1cm, minimum height=0.5cm,text centered, draw=black, fill=salmon] \tikzstyle{state} = [rectangle, minimum width=1cm, minimum height=0.5cm,text centered, draw=black, fill=darkblue] \tikzstyle{arrow} = [thick,-&gt;,&gt;=stealth]  \begin{tikzpicture}[thick,scale=0.8, every node/.style={scale=0.8},node distance=1.5cm]     \node (0) [state] {\textcolor{white}{feature extraction}};     \node (1) [cell, below of=0] {invertBotConE2\_f48\_k1\_b4\_noskip};     \node (2) [cell, below of=1] {invertBotConE2\_f16\_k3\_b4\_noskip};     \draw [arrow] (0) -- (1);     \draw [arrow] (1) -- (2);     \draw  (0.east)  [arrow]  edge [out=0, in=0, out looseness=4, in looseness=4]  (2.north); \end{tikzpicture}   0,2 are my nodes. When east is specified, it creates an arrow on node 0. If removed, it disappears. What if I would like to put 0.south as the start of the edge? "Yesterday" time specification shown when it was actually the day before yesterday Looking at , I see "asked yesterday" (which I really hate; I'd like to see an exact time specification rather than this vague one, especially in the list of questions). But it was not asked yesterday! When I hover over it to show the exact time, it shows it was actually the day before yesterday!  This snapshot was taken at 2012-03-31 15:00 UTC (plus or minus a few minutes):    Whatever different opinions you may have on the implementation, yesterday is a word with a certain meaning. Redefining it is not a good idea. At least rephrase this as "1 day ago" or something. In Java, are there clear rules on when to use each of access modifiers, namely the default (package private), public, protected and private, while making class and interface and dealing with inheritance? Does the English language have an official Academy? For some languages, there are academies that decide topics such as grammar and spelling of things, for example, for the Spanish language, there are 22 academies in 22 different countries, all making decisions on spanish grammar in their country. I would like to know if there is something that serves the same purpose for the English language. We have a SQL Server 2012 instance that contains a database with data for many product lines.   We are looking for a way to take subsets of this data into another database on the same instance.  This doesn't have to be continuous as we are looking for ways to make it easy to export backups of the subsets that we can send out to vendors (we have a company that loads this data into their own sql server to produce a product we sell, naturally its huge and we want to trim it down to just the product lines they work with on a regular enough basis) or use to load SQLite DB's for use with offline applications.  ProductDB - the superset of all product line data.  ProductLine1DB - A subset identical in structure to ProductDB, but with data only relevant to product line 1.  ProductLine2DB - A subset identical in structure to ProductDB, but with data only relevant to product line 2.  This can be done nightly, weekly, as often as we need to but not a continuous replication.  We don't want to add in distributor servers or even extra subscriber servers. We would like the replication to happen on the same server with the subscriber databases being on the same server.  Is this a good candidate for transactional replication?  Why or why not?  We obviously have other options to load this data, but it seems like this might be a good case to use something like transactional replication. I've heard its powerful but prone to issues when not setup optimally.  Note: I am not a DBA, simply a developer, but we are currently looking for a new DBA and some of us are taking over duties like this temporarily. What is difference between: "I have been married with (was married with, married) her for three years." What is the difference between the three sentences below?   I have been married with her for three years. I was married with her for three years. I married her for three years. There is a (very minor) typo in the Close Votes review audit failure message. The message is presented as follows:  STOP! Look and Listen. This was an audit, designed to see if you were paying attention. You didn't pass. There are no major problems with this question. You should click Leave Open, or make minor corrections via Edit. Don't worry, we've already handled this post appropriately – but please take a minute to look it over closely, keeping in mind the guidance above.;  The ; at the end is a typo and is in the actual message that was displayed. Where are the outliner toggles in 2.8? Can I have 2x2 workspaces in GNOME 3? I have just upgraded to Ubuntu 17.10 and found out there is no option to have 2 by 2 workspaces. I was able to configure 4 "static" workspaces using , but I would like to have them in 2 rows by 2. Any ideas? How prove this diophantine equation $x^2+y^2+z^3=n$ always have integer solution show that: For any postive ineteger $n$,then the equation    $$n=x^2+y^2+z^3$$   always have integer solution   My idea: such as $n=1$,then we have $$1=0^2+0^2+1^3$$ $$2=0^2+1^2+1^3$$ $$3=1^2+1^2+1^3$$ $$4=2^2+0^2+0^3$$ $$5=1^2+2^2+0^3$$ $$6=1^2+2^2+1^3$$ $$7=2^2+2^2+(-1)^3$$ $$8=0^2+0^2+2^3$$ $$9=1^2+0^2+2^3$$ $$10=1^2+1^2+2^3$$ $\cdots\cdots\cdots$  But for  general $n$, How prove it? Ways to obtain new portrait photography clients? I'd like to increase my portrait business. I've had good experiences with family, pet portraits and High School seniors.  My local newspaper is quite expensive. I've had okay luck with Craigslist, although Craiglist shoppers are far from my ideal client that is willing to pay for quality images, fast turn around and a great experience.  I have a flyer on my community message board, but not enough traffic walks by it. I was thinking about posting at my local churches. I get a few of referrals from friends and family of clients.   What are good ways to reach my local base and increase sales? I want to apply a macro to all the cells of a column with the cell content as the input to the macro. Is this possible?  For example if the cell contents have the word TEST, the macro should take TEST as its argument.