Let $\mathbb{F}[x]$ be a polynomial ring over a field $\mathbb{F}$. Let $I \subset \mathbb{F}[x]$ be an ideal, that is,  $\forall s,t \in I: s-t \in I$ and $\forall r \in \mathbb{F}[x]: rs,sr \in I$.  Prove that $\exists c(x) \in \mathbb{F}[x] : I = \{c(x)g(x): g(x) \in \mathbb{F}[x]\}$.  Any hints as to which $c(x) \in \mathbb{F}[x]$ I should pick and how I should proceed? Also my notes highlighted this result as particularly important... why is this? Thanks!
Let's start with a definition:     Definition: Let $F$ be a field, $I$ a nonzero ideal in $F[x]$, and $g(x)$ an element of $F[x]$. Then, $I= \langle g(x)\rangle$ if and only if $g(x)$ is a nonzero polynomial of minimum degree in $I$.   Now it gets completely foggy what is meant with "minimum degree".I constantly come across the "minimal polynomial". This seems to be linked, but it seems strange that there is a division here.   Could someone give a pointer to what the second line actually means. Maybe some example or diagram (if possible) would help a lot here.
When & how did Yoda come to know about the Rule of Two? At the very end of Star Wars Episode I: The Phantom Menace, Mace Windu and Yoda had this conversation:     Mace Windu: There is no doubt. The mysterious warrior was a Sith.   Yoda: Always two there are....no more...no less. A master and an   apprentice.   When did he come to know about the secret ? And, how? Any evidence from ? Why is pruning not needed for random forest trees? Breiman says that the trees are grown with out pruning. Why? I mean to say that there must be a solid reason why the trees in random forest are not pruned. On the other hand it is considered very important to prune a single decision tree to avoid over fitting. Is there some literature available to read for this reason? Of course the trees may not be correlated but there still will be a possibility of over fitting. Entire function and the series of its successive derivatives I recently came across the result that if $g$ is holomorphic in a neighbourhood of a point $z_0$ and  $$\sum_{n=0}^{\infty }{g^{(n)}(z_0)} $$ converges then $g $ is an entire function. Now, my doubt is if the above series is absolutely convergent then can we say something more about $g$, like say $g$ is a polynomial? I could not find any non-polynomial  entire function( I tried with functions like $e^z$, $\sin z $, et al), whose successive derivatives converge absolutely as series. Any help please? I commonly use the word 'guys' to refer to a group of males colloquially. It's colloquial but not rude, off putting, condescending, patronizing (though I wouldn't use it with a group of men at a board meeting (hypothetically of course), unless I knew them). So, some that I would not consider as replacements would be    buddies: too old-fashioned dudes: too informal boys: too patronizing men, gentlemen: too formal or false respect   I realized that I had also been using it for any group of people, males and females, even just females. It worked for me and I didn't think of the gender implications, that women might not care for it. No one ever complained, directly or indirectly (rule of life: sometimes people don't complain about things they should).   But it occurred to me (maybe by reading something that sparked realization) that the intended hearers might not all care for it.  What might be a female gendered or non-gendered version of 'guys'?  Some that I've considered don't feel right about (though these may be reasonable answers) are:   dolls: too old-fashioned babes: too informal  ladies: too formal or I've heard from women, too creepy women: too factual girls: too patronizing (as much as 'boys') everyone, you all: too bland y'all: perfectly non-sexist but too regional (might work, but not everywhere)   Hopefully I haven't eliminated all the possibilities -- any suggestions? How much set theory and logic should typical algebraists/analysts/geometers know? (soft-question) I know enough amount of set theory and logic to study grad-level math. However, I don't know more advanced set theory and logic, such as the ones on Kunen's or Shoenfield's texts. Although it's good to learn such advanced topics, I may not have enough time to do so, since I will probably put my emphasis on algebra, analysis or geometry as well as mathematical physics.   Although how much set theory and logic I should learn is totally up to what sort of analysis, algebra or geometry I will study, I want to know how much of them typical competitive algebraists/analysts/geometers know. If you are one of them, how much do you know? Let, $D=\{z\in \mathbb C:|z|&lt;1\}$. Which are correct?   there exists a holomorphic function $f:D \to D$ with $f(0)=0$ &amp; $f'(0)=2$. there exists a holomorphic function $f:D \to D$ with $f\left(\dfrac{3}{4}\right)=\dfrac{3}{4}$ &amp; $f'\left(\dfrac{2}{3}\right)=\dfrac{3}{4}$. there exists a holomorphic function $f:D \to D$ with $f\left(\dfrac{3}{4}\right)=-\dfrac{3}{4}$ &amp; $f'\left(\dfrac{3}{4}\right)=-\dfrac{3}{4}$ there exists a holomorphic function $f:D \to D$ with $f\left(\dfrac{1}{2}\right)=-\dfrac{1}{2}$ &amp; $f'\left(\dfrac{1}{4}\right)=1$.   With the help of Schwarz lemma &amp; its applications, we find that $(1)$ is false &amp; $(3)$ is true.  But, I can not think about options $(2)$ &amp; $(4)$. How can I speed up data transfer between local computers on a WiFi network? I have a router with ADSL Pentagram Cerberus 6331-42. Is it normal to have transfer speeds between two computers of max 1MB/s (not 1 Mbit)? How can I set my router up to make it faster at transferring data in my local WiFi network?  I have a PC and a laptop. Both are running Windows 7. I don't use a firewall. I've set the network area up correctly; I can see other computers.  Is it possible to make the router faster by changing its values? I don't use secure encryption such as WPA or WEP; only MAC filtering.  My values in Interface Setup > Wireless are:       Here are my thoughts after few years I've published this topic:    use 5Ghz bandwidth if possible  use range extender if Your router is too far  try wifi analyzer app for mobile phone to check what channel is the less used I'm trying to enable Intel SRT on my laptop. To do this, I need to change SATA controller mode from AHCI to RAID. The problem is that windows has no drivers for RAID and I can't install it while controller is in AHCI mode.   For now I have RAID driver in INF package (inf, sys, cat files). And I can load Windows recovery console with controller in RAID mode. The last thing to do is to intall this driver, but I don't know how to do it.  Google says rundll32.exe setupapi,InstallHinfSection DefaultInstall 123 &lt;filename&gt;.inf might help, but it doesn't. How to make an international emergency call? I am working with a basic  which is a popular modal used on many sites. I noticed that the default template comes with both a close button on the bottom, as well as an x on the top, these two buttons essentially having the same purpose. Are the two redundant and the close button completely unnecessary and just taking up extra space? Or, is having multiple usage options more user friendly?   To make this even more fun, suppose I throw in a Save button:   Is it now clearer that if you don't press save and just press the x, your changes will be canceled, or is there now more of a need for a cancel button for users who are scared that a mistake will become permanent? Update: To clarify for those who feel this question is similar to  - That question is asking if a clickable background is sufficient to close a popup. I'm using buttons and it is just a matter of how many buttons to use. (I do also have a clickable overlay and ESC key enabled, but that is irrelevant to my question) I'm looking to clarify the differences/overlaps between the three and, if the terminology isn't accurate establish the correct terminology Does someone know how to set a command block with a command inside using the setblock command?  I have created this code /setblock &lt;x&gt; &lt;y&gt; &lt;z&gt; command_block {TileEntityData:{Command:"testfor @p[_=1]"}} but it doesn't works.  Do someone know how to do that? Say I have two independent random variables $X \sim N(u_1, \sigma_1)$ and $Y \sim N(u_2, \sigma_2)$. I want to get the conditional distribution of X given whether X is bigger than Y or not.  $P(X|X&lt;Y)$ = ... and  $P(X|X&gt;Y)$ = ...  I am thinking solving this in this way:  \begin{align} P(X|X&lt;Y) &amp;= \frac{P(Y&gt;X|X)P(X)}{P(Y&gt;X)} \\ &amp;= \frac{(1-\Phi(\frac{x-\mu_2}{\sigma_2}))N(\mu_1,\sigma_1)}{\Phi(\frac{\mu_2-\mu_1}{\sqrt{\sigma_2^2+\sigma_1^2}})}\\ P(X|X&gt;Y) &amp;= \frac{P(Y&lt;X|X)P(X)}{P(Y&lt;X)} \\ &amp;= \frac{\Phi(\frac{x-\mu_2}{\sigma_2})N(\mu_1,\sigma_1)}{1-\Phi(\frac{\mu_2-\mu_1}{\sqrt{\sigma_2^2+\sigma_1^2}})} \end{align}  My questions are:   (1) Whether above solution is correct  (2) How to get the mean and sd for $P(X|X&lt;Y)$ and $P(X|X&gt;Y)$ if they are still normal? Convert list of ints to one number? I have a list of integers that I would like to convert to one number like:  numList = [1, 2, 3] num = magic(numList)  print num, type(num) &gt;&gt;&gt; 123, &lt;type 'int'&gt;   What is the best way to implement the magic function?  EDIT  I did find , but it seems like there has to be a better way. I was surprised to discover my dictionary had this entry for dilemma:     a situation in which a difficult choice has to be made between two or more alternatives, esp. equally undesirable ones   The notion of dilemma meaning two or more flies against what I was taught about the word. The very idea of a  is specifically based on the number two.  Has my dictionary merely updated its definition to encapsulate the many people who use dilemma for more than two equal choices? Or was someone in my youth being unnecessarily pedantic? How to make existing public files private? PHP MySQLi doesn't recognize login info Extended Rigid Bodies in Special Relativity I was reading Landau &amp; Lifshitz's Classical Theory of Fields and I noticed that they mention that an extended rigid body isn't "relativistically correct".   For example, if you consider a rigid rod and apply a torque at one end, by definition of being rigid, the whole body must start rotating at the same instant. But the information about the force being applied cannot travel faster than the speed of light.   The book hasn't mentioned how to resolve this paradox, and after thinking about it for a while, I'm wondering if one can even define a rigid body in a relativistically correct manner.  To summarize: How does one define a rigid body in special relativity? What was the point of testing carbonite-freezing technology on Han Solo? Darth Vader tested carbonite-freezing technology on Han Solo because he didn't want to kill Luke who was precious to the emperor. But, why was it necessary? Wasn't it already a tested and proven technology? Even Darth Vader (when he was Anakin) himself used it along with others decades ago (Star Wars: The Clone Wars S03E18). Why are certain example URLs like https://site.com/ and https://mysite.com/ blocked from post content?