Let $F$ be a field and $\bar{F}$ be its algebraic closure. Is it true that $|\bar{F}:F|=\infty$ implies that the degree of the irreducible polynomials in $F[x]$ is unbounded?     I believe it is, and in this case an hint in the right direction for a proof would be enough, but I don't know where to begin
One very convenient property of $\mathbb{R}$ as a ring is that there is an upper bound for the degree of irreducible polynomials in $\mathbb{R}[x]$, as      If $f\in\mathbb{R}[x]$ has degree larger than $2$, then $f$ is reducible.   However, the proof as I know depends highly on the fact that $\mathbb{C}$ is algebraically closed, and the very nice property: $z\in\mathbb{C}$ is in $\mathbb{R}$ if and only if $z=\bar{z}$.  This makes a generalization to other integral domains rather difficult. So the problem is      For what kind of integral domain $R$, we have a finite upper bound on the degree of irreducible elements in $R[x]$?   Some most familiar examples are ruled out: in $\mathbb{Z}[x]$, $\mathbb{Q}[x]$ and $\mathbb{F}_{p}[x]$, there is not such a bound. Unfortunately these exhaust all integral domains about which I have a working knowledge.  Another result might help is , which says if we can find a prime ideal $\mathfrak{p}$ in $R$ such that $\mathfrak{p}^2\neq\mathfrak{p}$, then by picking $a\in\mathfrak{p}^2\backslash\mathfrak{p}$ we have an irreducible $a+x^d$, where $d$ can be arbitrary, and hence the upper bound is not possible.  So we only need to focus on domains where $\mathfrak{p}^2=\mathfrak{p}$ for all prime ideals. This seems to be a quite strong restriction but I am not sure what to make of it.  Can someone give a hint?  Thanks!
Batch export Photoshop layers to individual PNG files When does reflow happen in a DOM environment? What kinds of activities will trigger reflow of web page with DOM?  It seems there are different points of view. According to ,  it happens   When you add or remove a DOM node. When you apply a style dynamically (such as element.style.width="10px"). When you retrieve a measurement that must be calculated, such as accessing offsetWidth, clientHeight, or any computed CSS value (via getComputedStyle() in DOM-compliant browsers or currentStyle in IE).   However, according to , taking measurement triggers reflow only when there is already reflow action queued.  Does anybody have any more ideas? In Java, is there any way to use the JDK libraries to discover the private classes implemented within another class? Or do I need so use something like asm? Why am I seeing a bomb icon on my files and folders in Ubuntu? Does borrowing from my 401(k) make sense in my specific circumstance? I know the standard answer and understand that I will be losing out on that compound interest, but I am young and only need to borrow a few thousand. here are some more details:   Need to borrow $6500 to pay off a car and two student loans ($300/month in payments combined) I am 22 and work full time, making about $70K annually I contribute 8% of my income with a 4% company match I increase that by 2% every year The loan would be over 48 months ($140/month) I live very frugally and have almost no discretionary spending that can be cut (don't eat out all the time, no gym membership/ Netflix/ Apple music etc.) I'm cash-poor: 90% of my salary is going to mortgage, groceries, students loans, car payment, newborn baby medical expenses, and commute. These are not my only debts, just the ones I am seeking to pay off in the short term. I work two jobs, one full and one part time. I started a business, and my wife has two jobs. Working more is not a good answer to the original question.   It seems to me that I would be reducing middle to high interest debt while also saving myself $150 per month.   Loans:     Student loan 1: $1200 @ 5%   Student loan 2: $1100 @ 4.25%   Car Loan: $4000 @ 6% 1970s or 80s kids book: girl's life taken over by a robot doppelganger version of herself I read this book between 1980-82 but it could have been written earlier.  It's about a girl who is dissatisfied with her life, as her parents have her on a diet and she wants to be able to eat what she wants. Someone offers her a solution: to come live in this large building where she can eat all the chocolate eclairs and other junkfood she wants, while a robot doppelganger android version of herself will take her place at home and at school.  The girl agrees, but finds that after week of eating mainly eclairs, she is sick and wants to eat a decent meal. But she is not allowed to eat anything else and has found herself to be a prisoner, so she sets out to get rid of the android and get her life back.   I don't recall the ending but it seemed the 'people' behind it all might have been aliens and the building might have been a kind of ship that then disappears. I have very little background in physics, so I apologize if this question is painfully naive.  Consider the following thought experiment: an observer is in a closed room whose walls, floor, and ceiling are made entirely of mirrors, with a single light source in the middle of the room. When the light is on, the observer can see many copies of his reflection all over the place.  Suddenly, the light source turns itself off. Intuitively, I would expect the observer to "instantly" see darkness. However, I can't figure out why that is the case under the "particle" interpretation of light. There are obviously lots of photons already in the room from before. Furthermore, we know that they don't get "consumed" when they hit a wall, because otherwise the observer wouldn't see so many reflections of himself. Basically, when the light goes off, what happens to the photons already in the room?  I suspect the answer goes something like this: the photons in the room lose a little bit of energy every time they bounce off a mirror, but it's so minuscule that we can still see more reflections than our eye can resolve anyway. When the light goes off, however, it takes them a very small fraction of a second to bounce around the room enough times to diffuse completely, which our eye cannot detect.  Is that about right? If we had a theoretical "perfect reflector", would the light remain trapped in the room forever? If we had instruments that could measure such things very finely, would it take (slightly) longer for the light to go out in a room made of mirrors as opposed to a room made of, say, black cloth? It has been some months since I've turned on my Playstation 3. Today, when I turned it on with my controllers, it didn't work. I thought "Hey, the battery surely ran out", so I tried to charge them but they don't seem to be charging at all. Could it be that they're officially weight papers by now? What's the main difference between int.Parse() and Convert.ToInt32 "Coming soon" pages & SEO I am setting up my personal website, and I have designed a simple "Coming Soon / Under Construction" page which basically has nothing but a small notice for the users.   What kind of SEO can I make on this page such that it won't harm the full website's ranking later on? Bearing in mind I am a layman - with no background in physics - please could someone explain what the "big deal" is with quantum entanglement?  I used to think I understood it - that 2 particles, say a light-year apart spatially, could affect each other physically, instantly.  Here I would understand the "big deal".  On further reading I've come to understand (maybe incorrectly) that the spatially separated particles may not affect each other, but in knowing one's properties you can infer the other's.  If that it the case, I don't see what the big deal is... 2 things have some properties set in correlation to each other at the point of entanglement, they are separated, measured, and found to have these properties...?  What am I missing? Is it that the particles properties are in an "un-set" state, and only when measured do they get set?  (i.e. the wave-function collapses).  If this is true - why do we think this instead of the more intuitive thought that the properties were set at an earlier time? Homotopy functions I'm new here. Can anyone give me a hint in this problem? Please.  Let $X$ be a topological space. If $f,g$ are continuous function of $f,g\colon X\to S^n$  which $f(x)\neq -g(x)$ for all $x$ in $X$, show $f$ and $g$ are homotopics. I try to show a homotopy using $$q(x) = \frac{f(x)+g(x)}{\|f(x)+g(x)\|}$$ it's well defined, but i'm stuck. Unable to see anything after importing .obj file Having trouble adding objects to NSMutableArray in Objective C Analytic continuation of function continuous on boundary Suppose one has a function $f$ in the disc algebra ie: $f$ is continuous on $|z|\leq1$ and holomorphic in $|z|&lt;1$. I wondered, can $f$ always be extended to a holomorphic function on some region containing the boundary (ie: $T$ the unit circle). If not can you give me an example? Thank you, Why is Facebook missing from the list of ways to share a page in Safari?  That option used to exist before the latest update.  When I'm directed to the extensions box Facebook is not there. How do you implement Coroutines in C++ get dynamic properties in javascript I can't seem to find an answer to this simple question, which I need for some compliance documentation.    On a default install of CentOS 6.5 (OpenSSH 5.3p1-94.el6), after how long of being idle will a user's SSH session be terminated?  I believe the following can be set to increase the idle timeout, but they are commented out by default.  $ grep -i alive /etc/ssh/sshd_config #TCPKeepAlive yes #ClientAliveInterval 0 #ClientAliveCountMax 3   Also, is there a command to dump a list of the current sshd settings?  I don't see anything in man sshd. What is the difference between a[bc]d and a{b,c}d? Why do people use a{b,c}d when there is already a[bc]d?