Let $k$ be a field and let $f\in k[x]$. Let's say that there are some $g, h\in k[x]$ such that $\gcd(g, h)=1$, $g|f$ and $h|f$. I want to show that $gh|f$.  Since $g|f$ and $h|f$, we may write $f=gu$ and $f=hv$ for some $u, v\in k[x]$. Now, $\gcd(g, h)=1$ implies that there are some $\alpha, \beta \in k[x]$ such that $\alpha g+\beta h=1$. We may now write that $gu=hv\implies \alpha gu=\alpha hv \implies (1-\beta h)u=\alpha h v\implies u=h(\alpha v +\beta u)$. Thus, $f=gh(\alpha v + \beta u)$, so $gh |f$. Is this correct? I remember hearing this fact in high school, but in my abstract algebra course from this year the lecturer never mentioned it, even though it would have made some things easier. This is why I tried to prove this myself.
I need your help with this: Let $f$, $p$, and $g$ be a polynomials in $F[x]$, assume that $f$, $g$, and $p$ are nononzero.  If $\gcd(f,p)=1$ and $fg$ is divisible by $p$ with no remainder, I need to prove that $g$ is  divisible by $p$ with no remainder  Thanks.
When is "EDIT"/"UPDATE" appropriate in a post? I often see questions edited into this form:     Lorem ipsum dolor sit amet blah blah blah paragraph paragraph original question.            UPDATE: The Lorem disappeared and left this message: "new information".   I find this format unaesthetic (since Stack Exchange posts are supposed to be useful in the long term and update boundaries are artifacts of the creation rather than the content) and feel that new information should, whenever possible, be incorporated into the original text “as if it had always been that way” if practical, and pointing out an update should be done mostly in comments.  Is this an appropriate writing style, and if so under what circumstances? If not, should it be removed or discouraged? Some Git commands take commit ranges and one valid syntax is to separate two commit names with two dots .., and another syntax uses three dots ....  What are the differences between the two? What do I call the ′ in mathematical formulae? As in x′ = x + t  "Ex (?) equals ex plus tee".  In Russian it is called "штрих" (shtrikch). Rufus nor any other software can fix my 16GB usb drive after using RUFUS I wanted to make a bootable USB drive using Rufus. I opened the program went with all the default options and I clicked start BUT what I forgot to do is to select an iso because I thought it would just make the disk bootable without selecting anything. Unfortunately it did not. It gave me an error. I do not remember it because this was about 6 months ago after I gave up in fixing it. Anyway I tried to reformat it again using diskpart and it fails. With Rufus it says "ERROR while partitioning drive" and I think that was the initial error as well. Also during the struggle I got messages from windows saying that the disk is write protected and NO, there is no switch for that on the USB pendrive.   More info:     Transcend 16 GB     Rufus 1.4.9,     Frustrated student :/ Below is an example from I.N. Herstein:       Let  $R=\Bigg\{\left( \begin{array}{ccc} a &amp; b \\ -b &amp; a \end{array} \right)\Bigg|a,b\in \mathbb R\Bigg\}$ and let $\mathbb C$ be the field of complex numbers. Define $\psi :R\to \mathbb C$ by $\psi\left( \begin{array}{ccc} a &amp; b \\ -b &amp; a \end{array} \right)=a+bi.$       It is asked to prove that $\psi$ is a ring isomorphism of $R$ onto $\mathbb C$ .    can anyone help me with some hint how to prove the ring isomorphism. TikZ/PGF 3.0 datavisualization vs pgfplots To follow-up on a previous question , the current release of TikZ 3.0 has a flexible datavisualization engine that creates the same kind of plots as the more mature plotting package pgfplots.  What is the current recommendation for producing complete plots in LaTeX? How to find and replace using sed on records starting with Specific number Is there a word that means "a span of six months"? That is, I want to connote a stretch of time that lasts half a year, not an event that happens every six months. How to Toggle Selected Objects Visibility with H shortcut as in Maya I want to be able to toggle the visibility of selected objects with a shortcut (H) the way you can do in Maya and XSI, shown in the GIF. I'm guessing scrippting is needed. I've tried to do it my self but I'm new to Python and Blender and as soon as the objects are hidden they seem to no longer be in a selection state/list. Eg if I manually select the hidden object/s in the Outliner and try to print the selected object to the console, none seem to be selected. Is this possible to do with the current state of Blender 2.8? Probability of a painted cube being reassembled into itself Suppose 27 cubes are stacked together, suspended in the air, to form a larger cube.  The cube is then painted on all the exposed surfaces and dried.  The smaller cubes are then randomly permuted in both spatial position and spatial orientation to form another large cube.  What is the probability that new larger cube is identical to the original?  This is not homework, it was passed on by a friend who found it somewhere on the internet, and they do not remember where. Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix Capitalize fields of study? Do I say "I study computer science," or "I study Computer Science"? Similarly, "I really liked that computer science course," vs. "I really liked that Computer Science course." Where does the extra kinetic energy come from in a gravitational slingshot? Troubleshooting "Windows cannot be installed to this disk. The selected disk has an MBR partition table" I recently bought a desktop computer with Windows 7 Home Premium x64 preloaded (OEM). I tried to install Windows 7 Ultimate x64 on this computer but I was presented with the following error message:     Windows cannot be installed to this disk. The selected disk has an MBR   partition table. On EFI systems, Windows can only be installed to GPT   disks.   The 1TB HDD on this computer has three partitions:   SYSTEM 100 NTFS - Healthy (System, Active Primary Partition)  OS (C:) 9148.04 GB NTFS - Healthy (Boot, Page File, Crash Dump, Primary Partition)  HP_RECOVERY (D:) 13.37 GB NTFS - Healthy (PrimarynPartition)   I'm trying to install a clean copy of Windows. This question is about cosmology and general relativity. I understand the difference between the universe and the observable universe. What I am not really clear about is what is meant when I read that the universe is infinite.   Does it have infinite mass or is it dishomogeneous?  How can the universe transition from being finite near the big bang and infinite 14 billion years later? Or would an infinite universe not necessarily have a big bang at all? LINQ Select Distinct with Anonymous Types So I have a collection of objects.  The exact type isn't important.  From it I want to extract all the unique pairs of a pair of particular properties, thusly:  myObjectCollection.Select(item=&gt;new                                 {                                      Alpha = item.propOne,                                      Bravo = item.propTwo                                 }                  ).Distinct();   So my question is:  Will Distinct in this case use the default object equals (which will be useless to me, since each object is new) or can it be told to do a different equals (in this case, equal values of Alpha and Bravo => equal instances)?  Is there any way to achieve that result, if this doesn't do it? I am looking for a story about the future of mankind, in which most people have been enslaved in a virtual reality. It ends with the revelation that the people are just lying there imagining their lives with their heads in a trough of feed.  It explains that chemists began to create compounds that could be released into the air to manipulate citizens - causing them to live, shop, think as desired by whoever was in charge. Eventually these tools got subtle enough that they could simulate reality using just these airborne compounds.  This was clearly a prescient view of what later became computer-based virtual reality fiction. Java - Splitting a CSV file into an Array String manipulation with & or + in VB.NET counting number of specific string in each row I have a SNP table looks like this  A N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N C C N N N N N N N N N N C C N N N C C N N N N N N N N N N N N N N N N N   I want to count total number of N in each row? any idea how to do that