I am trying to prove:     Let $K$ be the splitting field of $f(x)=x^4+ax^2+b$, whose roots are denoted by $\pm\alpha,\ \pm\beta$. Prove that $Gal(K/\mathbb Q)\cong V_4$ if and only if $\alpha\beta\in\mathbb Q$.   My question is: if we let $$\sigma:\alpha\mapsto-\alpha,\ \beta\mapsto\beta$$ $$\tau:\alpha\mapsto\alpha,\ \beta\mapsto-\beta$$ Then if both are well-defined automorphisms of $K$, the group $$\{1,\sigma,\tau,\sigma\tau\}$$ is isomorphic to $V_4$. However, $\sigma,\tau$ do not fix $\alpha\beta$, which means that this group must not be $Gal(K/\mathbb Q)$ (otherwise $\alpha\beta\notin\mathbb Q$, contradictory to what I am trying to prove) What is the reason?  p.s. I am aware that there questions asking about the same proposition. But they don't mention the problem that I am having here.
I want to compute the Galois group of some polynomials, but I want to see some examples first. For example this proposition could be helpful. I don't know how to prove it &lt;.&lt;  Let's consider a polynomial $f(x)= x^4+ax^2+b \in \mathbb{Z} $. Let $ \pm\alpha , \pm\beta$ denote the roots of $f(x)$. Note that $f$ is irreducible iff $\alpha^2, \alpha\pm\beta \notin \mathbb{Q}$. So let's suppose that $f(x)$ is irreducible. Let's denote $G$ the galois group of $f$ (the Galois group of the splitting field $\mathbb{Q}(\alpha,\beta)/\mathbb{Q}$)  $i)$ $G\cong V$ the Klein 4-group iff $\alpha\beta \in \mathbb{Q}$  $ii)$ $G\cong \mathbb{Z}_4$ iff $\mathbb{Q}(\alpha\beta)=\mathbb{Q}(\alpha^2)$  $iii)$ $G\cong D_8$ the dihedral group of order 8, iff $\alpha\beta \notin \mathbb{Q}(\alpha^2)$  I need help with the proofs of this propositions... Well I'm very lost with the computation of Galois group, some care must be exercised, I don't have it. For example I know that Galois group acts transitively, in the sense that $u\in K/F$ a galois extension, the other roots of the minimal polynomial of $m_u(x)\in F[x]$ are precisely the elements $ \sigma(u)$ , where $\sigma \in Gal(K/F)$. And clearly any automorphism, is determined by only the value of a basis. In this case we are working on $\mathbb{Q}(\alpha,\beta)$, so I have to know the values of both. But I have to respect the algebraic relations involving $\alpha , \beta$ I proved that in $i),ii)$ $\mathbb{Q}(\alpha,\beta) = \mathbb{Q}(\alpha) $ So I have to know the values on $\alpha$.   My first question is: If in both cases $i),ii)$, I have an extension of degree 4 (so $|G|=4$) I have to map $\alpha$ to determine the automorphism, but I have also to map $\alpha$ to all of it's four roots. Why the galois group in this case are different?  My second question is about $iii)$ I have no idea How to attack it.
When will my device get the Android 4.2 update (Jelly Bean)? How do you return to the default view (the one you see when blender starts)? Are punches with Gauntlets of Ogre Strength magical, improvised weapon attacks? My Gnoll Paladin just got the Gauntlets of Ogre Strength from Phandelver the other day, and it got me wondering:    Would it be possible/make sense for punches using them to count as an improvised weapon? (Since I'm not punching with empty fists, I'm wearing magical gauntlets) If so, would punches from a person using them count as magical weapon damage? How do I transfer data and settings from one Apple Watch to another? I have an Apple Watch. I have had it for a while. I just bought a new Apple Watch. (Both are Series 1, just different sizes.)  I would like to transfer all my data and settings from my old Apple Watch to my new Apple Watch and then sell my old Apple Watch.  Is it possible to migrate my profile from one Apple Watch to another?  If so, what is the procedure?  All the search results I have found have to do with moving an Apple Watch to a new iPhone or restoring a backup to the same Apple Watch. Perhaps it is possible to restore a backup from an old Apple Watch onto a new one? Understand PAM and NSS In the last days I have set up some Linux system with LDAP authentication and everything works fine, but there's still something I can't really understand regarding NSS and PAM, also after a lot of research. Citing:  NSS allows administrators to specify a list of sources where authentication files, host names and other information will be stored and searched for  and  PAM is a set of libraries that provide a configurable authentication platform for applications and the underlying operating system  What I don't understand is how PAM and NSS work and interact together. In  the architecture is explained pretty well: I configure PAM to use pam_ldap for LDAP accounts and pam_unix for local accounts, then I configure nsswitch.conf to fetch information from local files and LDAP. If I have understood correctly LDAP is used twice: first by pam_ldap and then by NSS which is itself called from pam_unix. Is that right? Is LDAP really used twice? But why do I need to configure both NSS and PAM? My explanation is that PAM performs different tasks than NSS and it is used by other programs. But, then, it should be possible to use only NSS or only PAM, as I have read in . So I experimented a bit and I have first tried to delete LDAP from the nsswitch.conf (and the authentication stopped to work as if only pam_ldap is not enough to do the job). Then I re-enabled LDAP in NSS and I deleted it from the PAM configuration (this time everything worked fine, as if pam_ldap is useless and NSS is enough to authenticate a user). Is there anyone who can help me to clarify this? Many thanks in advance. UPDATE I've just tried something now. I removed again all the pam_ldap entries in all pam configuration fields and I have also removed shadow: ldap from nsswitch.conf. As now in all the system there are only the lines: passwd: ldap files and group: ldap files in nsswitch.conf. Well... the login with LDAP users works perfectly, those two lines (plus /etc/ldap.conf) are enough to configure LDAP auth. From my knowledge PAM in independent from NSS, but my tests showed it's not. So I ask myself is it possible to completely disable NSS and use only PAM? How do I cure Sanguinare Vampiris? I have Sanguinare Vampiris. I don't really want to become a vampire. How can I cure myself? Can I, as a reviewer, tell authors to get a native English speaker to proofread their paper? I am reviewing a paper for possible publication in a respected journal. The English in the paper is very poor. The authors are clearly not native English speakers. I want to write something to the effect of the following in my review.   I advise the authors to find a native English speaker to proofread the manuscript.    My question: Is this appropriate in a review?  On one hand, I think it is good, constructive advice. The paper would be significantly clearer if someone spent a few hours helping them fix it up. I can try to help them through reviewer comments, but it would be much easier if someone could help them in person. The authors are located in a western English-speaking country, so they should be able to find someone.  On the other hand, I don't want to be "the mean reviewer." I understand that English can be difficult to master for immigrants. Perhaps there is a more diplomatic way of saying this. Last two digits of $2^{1000}$ via Chinese Remainder Theorem? I bumped across the aforementioned question in my notes while studying today and I have completely forgot how to do this. I remember using CRT to solve a problem like this on one of my tests, too bad they didn't give back my solutions :(.  Since $\gcd(100,2) = 2$, we can't use the usual Euler's theorem to solve via $\pmod {100}$. So $100 = 5^2 2^2$, and applying Euler's on the 25 gives me $2^{20} \equiv 1 \pmod {25}$. However since $\gcd(2,4) = 2$, we can't do the same for $\bmod 4$. Accordingly how do I set up the other modulo congruence so I can apply CRT? These days, the standard way to present differential calculus is by introducing the Cauchy-Weierstrass definition of the limit. One then defines the derivative as a limit, proves results like the Leibniz and chain rules, and uses this machinery to differentiate some simple functions such as polynomials. The purpose of my question is to see what creative alternatives people can describe to this approach. The nature of the question is that there is not going to be a single best answer. I have several methods that I've collected which I'll put in as answers to my own question.  It's not reasonable to expect answers to include an entire introductory textbook treatment of differentiation, nor would anyone want to read answers that were that lengthy. A sketch is fine. Lack of rigor is fine. Well known notation and terminology can be assumed. It would be nice to develop things to the point where one can differentiate a polynomial, since that would help to illustrate how your method works and demonstrate that it's usable. For this purpose, it suffices to prove that if $n&gt;0$ is an integer, the derivative of $x^n$ equals $0$ at $0$ and equals $n$ at $1$; the result at other nonzero values of $x$ follows by scaling. Doing this for $n=2$ is fine if the generalization to $n&gt;2$ is obvious. Salesforce to Xero integration using Named Credentials by OAuth 2.0 What is an observer in quantum mechanics? My question is not about (pseudo) philosophical debate; it concerns mathematical operations and experimental facts.  What is an observer? What are the conditions required to be qualified of observer, both mathematically and experimentally? Uniform vs discrete Gaussian sampling in Ring learning with errors The Wikipedia article on  mentions two methods of sampling "small" polynomials namely uniform sampling and discrete Gaussian sampling. Uniform sampling is clearly the simplest, involving simply uniformly selecting the coefficients from the set of small coefficients, thereby guaranteeing they will be small. However, the articles mention that security proofs rely on discrete Gaussian sampling.  However, I have a bit of difficulty wrapping my head around    Why discrete Gaussian sampling would be better than uniform, security-wise?   I mean, usually uniform is preferable - it maximizes the entropy/information in each sample, making the outcome more unpredictable. I could understand it if the Gaussian sampling technique would somehow serve to avoid (at least with high probability) "weak" parts of the space of outcomes but I have found no information/justification that this is the case.  Another reason I can see why discrete Gaussian might be better is that there is, after all, a chance (albeit a small one) one would select a non-small coefficient, so the space of outcomes is actually larger. However, given that this probability of hitting this extra space is (by design) quite small but also that one would likely immediately reject such a sample (because the algorithm calls for a small polynomial), this seems to be irrelevant.  It is not because I am against discrete Gaussian sampling I just want to understand why it is more secure. Also, I want to understand if it really IS more secure or it is simply easier to prove security in that model. I would actually consider that very interesting if it is the case because it raises the question of why is it easier to prove the security under conditions that are intuitively worse (i.e. discrete Gaussian vs the intuitively better uniform case)? If it is so intuitively clear that uniform sampling is better, surely it must be possible to transform that intuition into a security proof of RWE also under uniform sampling. Otherwise, I would fear the intuition must with high likelihood be wrong or at least dubious. Missing entry in flagging history I happened to glance upon my flagging history and I realized that there was an entry missing from the spam section in it. I had marked 6 posts as spam, out of which 4 were deemed helpful and 1 was declined. I'm not sure what happened to the remaining 1 entry and why it is missing from the history summary. Here is a snapshot of the spam section of my flagging history.    Is this behavior a  or a ?  Note: I've checked my flagging history completely and there is no spam flag which is still in the active state. I just installed Ubuntu 16.10 on my desktop and have a USB deathadder. The problem is I can't see where the cursor actually is because it appears only in the top left of my screen. The mouse still moves but the cursor graphic doesn't follow set of almost complex structures on $\mathbb R^4$ as two disjoint spheres The set of almost complex structures on $\mathbb R^{2n}$ is given by $$ M_n = \frac{GL(2n,\mathbb R)}{GL(n,\mathbb C)} = \mathcal C_+ \sqcup \mathcal C_-,$$ taking into account that $\det = \pm 1$ gives two disjoint sets.  How do we show that $M_2=S^2 \sqcup S^2$? Moreover, I'm a bit confused by dimensions: how does $\dim_{\mathbb R}M_2=8$ relates with the latter decomposition?  This is exercise 1.2.1 from the book Complex geometry by Huybrechts. How can I give other drives and partitions short, meaningful names in nautilus? In nautilus and Unity the external drive has a nice short descriptive name "mcat", but all partitions on the 4 internal drives are displayed as a size (834 GB filesystem) or a huge 32-character string form of a GUID:  I'm guessing the external drive is nice, short, sweet and readable because that drive may have no partitions (well, just one I guess) and that name may be the drive label, whereas partitions usually don't have names. That may explain my problem, but doesn't solve it.  Is there some way to give reasonable names to these partitions in nautilus and Unity? In a semiconductor memory chip, each bit of binary data is stored in a tiny circuit called a memory cell consisting of one to several transistors. Volatile type.   Suppose an application stored its data in a particular segment on the computers RAM.  How would the CPU know what data to extract and how would it sort through the other data to get to it. If everything in its core level is 0 or 1 its hard to distinguish what purpose 'x' memory has. Who uses Dual_EC_DRBG? Recent news articles have suggested that the NSA may be involved in trying to influence the cryptography in public standards or commercially deployed software, to enable the NSA to decrypt the encrypted traffic.  For example, .  The New York Times article specifically mentions a 2006 NIST standard, where two Microsoft engineers later discovered a serious problem in the scheme.  The New York Times article explains how the standard was written by the NSA, who pushed the scheme hard:     Simultaneously, the N.S.A. has been deliberately weakening the international encryption standards adopted by developers. One goal in the agency’s 2013 budget request was to “influence policies, standards and specifications for commercial public key technologies,” the most common encryption method.      Cryptographers have long suspected that the agency planted vulnerabilities in a standard adopted in 2006 by the National Institute of Standards and Technology, the United States’ encryption standards body, and later by the International Organization for Standardization, which has 163 countries as members.      Classified N.S.A. memos appear to confirm that the fatal weakness, discovered by two Microsoft cryptographers in 2007, was engineered by the agency. The N.S.A. wrote the standard and aggressively pushed it on the international group, privately calling the effort “a challenge in finesse.”    After some sleuthing, I'm pretty certain this is a reference to the  pseudorandom number generator scheme described in NIST SP 800-90.  The weakness is that Dual_EC_DRBG appears to contain a backdoor, and anyone who knows the backdoor can totally break the PRNG.  The weakness was first described in  and was subsequently .  As a consequence, there are now good reasons to suspect that Dual_EC_DRBG contains a NSA backdoor, and the NSA might be able to spy on anyone who is using Dual_EC_DRBG.  My questions:   Who uses Dual_EC_DRBG? Do any commercial products use Dual_EC_DRBG? Does anyone use it? How do you explain cubic growth of a function When reading scientific papers I have seen people explain the growth of a variable linearly, exponentially. However how would one say for a variable which grows in quadratic fashion, or cubic?  I don't remember reading something like: "y grows cubically with x"  Can I say, y grows cubically with x?  What would be the proper way to explain cubic or quadratic growth of a variable as a function of another one?  How would you explain the given lines: For example, in the past, links to www.example.com wouldn't have meant Google would pass any value to say, forum.example.com. What is the current status of this? (Please provide a reference if possible).