Finding nth roots of complex number
The $n$ complex $n$th roots of a complex number $z$
Finding Zero Divisors and Units of Rings $\mathbb{Z}_n \times \mathbb{Z}_m$ How do I exactly find the zero divisors and units of a ring in the form of $\mathbb{Z}_n \times \mathbb{Z}_m$? For instance, how does one calculate zero divisors and units of $\mathbb{Z}_6 \times \mathbb{Z}_2$? I understand that a zero divisor is a non-zero element of $R$ such that $rs = 0$ for some other non-zero element $s$ in $R$. But I don't really quite understand how to apply that to a ring of the form $\mathbb{Z}_n \times \mathbb{Z}_m$. Also, the units of $\mathbb{Z}_6 \times \mathbb{Z}_2$ would be elements that send some element $s$ in $\mathbb{Z}_n \times \mathbb{Z}_m$ to the multiplicative identity, right? (which would be 1)? How do I apply these concepts to "product" rings? Thank you. The official Checkpoint out command line tool from CheckPoint, for setting up a SSL Network Extender VPN is not longer working from the Linux command line. It is also no longer actively supported by CheckPoint.  However, there is a promising  project, that tries to replicate the Java applet for authentication, that talks with the snx command line utility, called snxconnect.   I was trying to put snxconnect text utility to work in Debian Buster, doing:  sudo pip install snxvpn   and  export PYTHONHTTPSVERIFY=0 snxconnect -H checkpoint.hostname -U USER    However, it was mostly dying either with an HTTP error of:   HTTP/1.1 301 Moved Permanently:   or:  Got HTTP response: HTTP/1.1 302 Found   or:  Unexpected response, try again.   What to do about it?  PS. The EndPoint Security VPN official client is working well both in a Mac High Sierra and Windows 10 Pro. I have a MacBook Pro Retina (2015). After updating to OS Mojave, Bluetooth has stopped working. I get an error message that "No Bluetooth Hardware Found".  I have already tried to do the following:   Go to the /Library/Preferences and delete the com.apple.Bluetooth.plist file Successfully resetting SMC. (pressing Shift+ctrl+alt + power button) Removing the Bluetooth icon from the menu bar    None of the above solutions has worked for me. Remove Meta Privilege for MSE (only) What I am asking here is to please remove the privilege to participate in Meta for MSE.  This privilege is literally useless. You can't really have a meta site for a meta site. There's no Meta.Meta.SE here or any other SE site out there: Meta.SE covers for every site in SE, including itself.     When users find out that you don't really get anything new, it's kinda sad for those people. There is no use for it except just to be there and fill up the space between the first and third privilege. Privileges should be shown only if you earn something new, and this privilege to participate in the meta isn't new because it's where you report problems in all of SE, including Meta.SE.    So please remove the privilege to participate in meta just for Meta.SE.    This is not a dupe as this is a feature-request to remove it, not support on why it is there. Brokerage accounts are seen as relatively tax-inefficient, since you invest with post-tax dollars, and growth is taxed upon realizing gains. The traditional IRA allows you to deduct contributions and defer taxes until you take distributions in retirement ...  Unless you are covered by a retirement plan from work and make over   $71,000/year if single, or $118,000/year if married filing jointly.   The numbers above come from . Above those thresholds, you can still make contributions to a traditional IRA, but those contributions will be post-tax. Distributions are taxed as ordinary income, excluding post-tax principal.  Similarly, eligibility to make direct Roth IRA contributions phases out to zero according to a :   $131,000 if single, or $193,000 if married filing jointly.   Looks to me that someone who cannot make direct Roth IRA contributions also cannot deduct traditional IRA deposits from income.  Assume   investment period is 20 years, is some garden variety ETF (VTI/VXUS/etc), those investments return 5%, the current marginal tax rate is 28%, the expected marginal tax rate at retirement is 25%, the long term capital gains rate is 15%, the entity is not eligible to make Roth IRA contributions, and 401k contributions are already maxed out.   A $5000 of gross income contribution into a traditional IRA at retirement might grow to an after-tax sum of:  (5000) * (1 - 0.28) * (1 + (1.05 ** 20 - 1) * (1 - 0.25)) = $8064   Instead, since contributions were going to be taxed anyway, why not put it in a brokerage account? Then you could claim the favorable long-term capital gains rate.  (5000) * (1 - 0.28) * (1 + (1.05 ** 20 - 1) * (1 - 0.15)) = $8659   All I found was , and the accepted answer says to still make a traditional IRA contribution, but I still don't see why that would be a good idea. What am I missing here? It's well seen throughout the series that there is a “curse” on the DADA teacher position, that no one can hold that post for more than a year. But it seems like Quirrell has been around for longer than a year. What's up with that? What is currently the best way to get a favicon to display in all browsers that currently support it?  Please include:   Which image formats are supported by which browsers. Which lines are needed in what places for the various browsers. Roots of polynomials in field of prime characteristic p Let $F$ be a field of characteristic $p$, and let $\alpha \in F$. Let $f \in \mathbb{Z}_p[x]$ be such that $f(\alpha) = 0$. Apparently, we have $f(\alpha^p) = 0$. This was mentioned but not proved in my lectures. Can someone explain why please?  Thanks. "wait more than" vs. "wait FOR more than" Complex number isomorphic to certain $2\times 2$ matrices? I have been trying to prove this, but I am having trouble understanding how to prove the following mapping I found is injective and surjective. Just as a side note, I am trying to show the complex ring is isomorphic to special $2\times2$ matrices in regard to matrix multiplication and addition. Showing these hold is simple enough.  $$\phi:a+bi \rightarrow \begin{pmatrix} a &amp; -b \\ b &amp; a \end{pmatrix}$$  This is what I have so far:  Injective: I am also confused over the fact that there are two operations, and in turn two neutral elements (1 and 0). Showing that the kernel is trivial is usually the way I go about proving whether a mapping is injective, but I just can't grasp this.  $$ \phi(z_1) = \phi(z_2) \implies \phi(z_1)\phi(z_2)^{-1} = I = \phi(z_1)\phi(z_2^{-1}) = \phi(z_2)\phi(z_2^{-1}).$$  So if we can just show that the kernel of $\phi$ is trivial, then it also shows that $z_1 = z_2$. The only complex number that maps to the identity matrix is one where $a = 1$ and $b = 0$, $a + bi = 1 + 0i = 1$.  Using a similar argument for addition we can just say that the only complex number $z$ such that $\phi(z) = 0\text{-matrix}$, is one where $a=0$ and $b=0$, $a+bi=0+0i=0$.   Surjective:  I forgot to add this before I posted, but I honestly don't really understand how to prove this because it just seems so obvious. All possible $2\times2$ matrices of that form have a complex representation because the complex number can always be identified by its real parts and since the elements of the $2\times2$ matrix are real then the mapping is obviously onto.  I have always had trouble understanding when I can say that I have "rigorously" proved something, so any help would be appreciated! $\newcommand{\P}{\mathbb{P}}$Is it valid to interpret $\P(B|A)$ as "infinite probability" when $\P(A) = 0$? Windows 7 / 8 stop modal dialogue flash on parent focus Is there some setting in Windows where I can change the default behavior that causes the border and title of a modal dialogue box to flash like 6 or 7 times when I click on the parent window?  Simply focusing it would be plenty to let me know what I did was wrong.   I don't feel the need for all the sirens and bull horns and strobe lights to let me know I clicked in the wrong place. As an example, consider the two sentences:     There don't seem to be any doctors here.   and     There doesn't seem to be any doctors here.   To my ear, the first sounds great, and the second is painfully awkward.  So which is correct, grammatically? I've found lots of disagreement on this around the Web, with various sources citing different ways of treating the word "any" (as singular, always, or as either depending on to what it refers). No consensus, however, could I locate. Floating Objects and Weight The Situation:  A ball is placed in a beaker filled with water and floats. It is also attached to the bottom of the beaker via a string.   The Question:  The ball is attached to the beaker, thus making the ball and beaker (and string included) a system. However, it is floating. Does the ball add to the weight of the beaker? Too many blanks between summation symbol and its associated expression Permanently turn off new Google Maps and go back to old one I found new Google Maps slow, useless (routes with only two points) and wrong. Is there any way, I can turn it off completely and go back to favored old Google Maps? Or is this &mdash; as with everything in Google &mdash; one way trip?  I found  text on this matter, but it seems useless. I have the feeling, that it discuss something completely different. It says about lite indicator in bottom-right corner. I don't have something like that. And provided links, that should switch new Google Maps engine between WebGL and canvas-based, simply doesn't work. All of them opens new window with annoying advertisment about new maps.  I even tried pasing part of URL, they're talking about (force=canvas) into any URL that my new Google Maps uses. Of course without any luck.  So... is there any way, I can get rid of new Google Maps, once and permanently?  UPDATE: In these days (April, 2015) Google , that soon new Google Maps will be the only one available. To my extremely negative surprise, this question will become invalid soon! :( I have a computer (Asus T200TA) with 2G RAM and 32-bit UEFI [Yep, what a wonderful idea to put 32-bit UEFI!]. I have two options:   Install a 64-bit system, boot manually and install grub-efi-ia32. Drawback: 2G is not much, systems with such low RAM are said to run better on 32-bit systems. Install a 32-bit version. (but from my state of knowledge it's not trivial to boot)   Which option would be better? This is a 2015 IMO problem.  It seems difficult to solve.     Find all triples of positive integers $(a, b, c)$ such that each of the numbers $ab-c$, $bc-a$, and $ca-b$ is a power of $2$.   Four such triples are $(a,b,c)=(2,2,2),(2,3,2),(3,5,7),(2,6,11)$. Constructing a function that is continuous and has a max on an open interval, but is not necessarily increasing immediately to the left of the max. Never delete entries? Good idea? Usual?