I need to show that the ideal generated by $x^2 − x + 1$ and 13 in $\mathbb{Z}[x]$ is not maximal.   I think that I need to check first that the polynomial is irreducible, but not sure how to follow from that.
Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$.  My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this?  $(0)$.  $(f(X))$, where $f(X)$ is an irreducible polynomial.  $(p)$, where $p$ is a prime number.  $(p, f(X))$, where $p$ is a prime number and $f(X)$ is an irreducible polynomial modulo $p$.
How does bash interpret the equal operator with no surrounding spaces in a conditional? The following script does not behave as I would have expected.  Adding spaces around the '=' in the conditional made it perform how I wanted, but it got me thinking, what is it actually doing inside the conditional?  #!/bin/bash S1='foo' S2='bar' if [ $S1=$S2 ]; then     echo "S1('$S1') is equal to S2('$S2') fi echo $S1 echo $S2   The output is:  S1('foo') is equal to S2('bar') foo bar   The contents of S1 and S2 don't change from what they are assigned, so the = doesn't perform an assignment. Word for giving animal characteristics (esp. physical ones) to humans I am writing a paper about Art Spiegelman's Maus, specifically the metaphor that Spiegelman creates by depicting his obviously quite human characters with the heads of various animals, or a couple times animal masks. The metaphor is used at times to bestialize/animalize the acts of the Nazis and, in some cases, to show the arbitrary nature of the classes/groups/divisions they and others applied to human beings.  I considered "chremamorphism," but this describes the attribution of the characteristics of an object to, or the objectification of, a human. I also considered "animalize" and "bestialize," but those describe some of the effects that can be produced by Spiegelman's metaphor, not the metaphor itself - in many scenes in the book, the characters are extremely human, and I think that the negative or degrading connotations of these words makes them inappropriate for this use. Nor do "anthropomorphism" or "personification" fit, because Spiegelman is not applying human characteristics to animals but the other way around. None of the words that I can think of quite fit.  What is a word that accurately describes Art Spiegelman's depiction of obviously human characters as having animal body parts or wearing animal masks in the comic book Maus? Why is list initialization (using curly braces) better than the alternatives? Calculating MEAN and STANDARD DEVIATION confussed [] ** calculate the mean and standard deviation of each investment, assume that both returns come from independent samples, are both normally distributed and that the (true but unknown) variances are the same,, Can someone could check my  calculations for  mean and standard deviation? I did calculations but not sure what to do with negative values of distances(If if am correct Investment A MEAN= 78.6875 when it comes to calculating standard deviation i have to power difference by 2 ?  ** Create unmanaged c++ object in c# Do singularities have a "real" as opposed to mathematical or idealized existence? I was thinking of, for example a Schwarzchild metric at r=0, i.e. the gravitational singularity, a point of infinite density. I realise that there are different types of singularities--timelike, spacelike, co-ordinate singularities etc. In a short discussion with Lubos, I was a bit surprised when I assumed they are idealized and I believe he feels they exist. I am not a string theorist, so am not familiar with how singularities are dealt with in it.  In GR, I know the Penrose-Hawking singularity theorems, but I also know that Hawking has introduced his no-boundary, imaginary time model for the Big Bang, eliminating the need for that singularity. Are cosmic strings and other topological defects singularities or approximations of them (if they exist).  In what sense does a singularity exist in our universe? --as a real entity, as a mathematical or asymptotic idealization, as a pathology in equations to be renormalized or otherwise ignored, as not real as in LQG, or as real in Max Tegmark's over-the-top "all mathematical structures are real"? Will I need any Malaysian Rinngit at Kuala Lumpur airport when I'm leaving the country? Does a family of a distribution have a different definition for statistics than in other disciplines?  In general, a  is a set of curves, each of which is given by a function or parametrization in which one or more of the parameters is varied. Such families are used, for example, to .  For statistics, a family according to  is the result of varying the shape parameter. How then can we understand that the gamma distribution has a shape and scale parameter and only the generalized gamma distribution has, in addition, a location parameter? Does that make the family the result of varying the location parameter? According to @whuber the meaning of a family is implicitly A "parameterization" of a family is a continuous map from a subset of ℝ$^n$, with its usual topology, into the space of distributions, whose image is that family.  What, in simple language, is a family for statistical distributions?  A question about relations among of the statistical properties of distributions from the same family has already generated considerable controversy for a  so it seems worthwhile to explore the meaning.  That this is not necessarily a simple question is born out by its use in the phrase , which has nothing to do with a family of curves, but is related to changing the form of the PDF of a distribution by reparameterization not only of parameters, but also substitution of functions of independent random variables. What is the difference between linear regression on y with x and x with y? The Pearson correlation coefficient of x and y is the same, whether you compute pearson(x, y) or pearson(y, x). This suggests that doing a linear regression of y given x or x given y should be the same, but I don't think that's the case.   Can someone shed light on when the relationship is not symmetric, and how that relates to the Pearson correlation coefficient (which I always think of as summarizing the best fit line)? Router to router setup works fine with straight cable up to 2-3 meter it is not working for 45 meter why? Motley fool says you can make $15,978 more per year with Social Security. Is this for real? The Motley fool has an ad campaign that says you can make $15,978 more per year on Social Security by doing 17 minutes of work using a legal loophole.  I don't want to link to their ad (although, you can find it by searching ).  What are they talking about here?  Is there any truth to this?  If someone could explain it to me without me having to watch the entire ad video or subscribe to their newsletter, I'd appreciate it. Confidence Interval for variance given one observation This is a problem from the "7th Kolmogorov Student Olympiad in Probability Theory":     Given one observation $X$ from a $\operatorname{Normal}(\mu,\sigma^2)$ distribution with both parameters unknown, give a confidence interval for $\sigma^2$ with a confidence level of at least 99%.   It seems to me that this should be impossible. I have the solution, but haven't read it yet. Any thoughts?  I'll post the solution in a couple days.  [Follow-up edit: Official solution posted below. Cardinal's solution is longer, but gives a better confidence interval. Thanks also to Max and Glen_b for their input.] tl;dr : For SEO, should I redirect from one to the other or list both on the DNS records?    It seems that most experts think that both www and non-www should exist. (Please correct me if I'm wrong.) And Google Webmasters Tools lets you choose which is the main one.  But: will a redirect give all SEO weight to one of them instead of splitting it? (There are other search engines after all, so setting the main one in Google Webmasters Tools might not be enough.) Will having both on the DNS records give them more importance? No difference? I have two custom taxonomies applied to two custom post types. the terms list on the sidebar just fine and will list all posts associated with it. However, if you search one of the terms in specific, it doesn't bring up a post with that term.  Example:  Search for term "PQRI"  I get nothing. Any ideas? I've tried using various search plugins but they either break my custom search parameters or just don't work. I have this HW where I have to calculate the $74$th derivative of $f(x)=\ln(1+x)\arctan(x)$ at $x=0$. And it made me think, maybe I can say (about $\arctan(x)$ at $x=0$) that there is no limit for the second derivative, therefore, there are no derivatives of degree grater then $2$. Am I right? If "No One" uses a face from the Hall of Faces to become someone then how did he do this Do we need inertial frames in Lagrangian mechanics? Consider a set $A$ of $10$ integers, $A = \{a_1,a_2, \ldots ,a_{10}\}$. Prove that there is at least one subset of $A$ whose sum is divisible by $10$. I'm trying to remember a cartoon tv show where the main boy, a blonde girl, and another boy capture monsters. There was something about a 'veil' between their world and the monster world, but a 'secret society' of sorts stopped the monsters from crossing, or used them to battle the people who wanted to use them for evil led by a wise old lady. I'm pretty sure they trapped monsters into pieces of jewelry or clothing.   Not so sure about this part: and they had these 'supernatural powers' that were connected to their 'aura' and they could travel from place to place with it.   (Another brown haired girl joined because she found out? There was an arc where the blonde got jealous and the monster she controlled attacked the brown haired girl?)  I can't remember much of it, but I remember in the end the 'veil' was broken and monsters flew over the city. And the main boy and the blonde girl held hands... or kissed? UTF-8 all the way through