How to prove that $\mathbb{Z}[X]/g\mathbb{Z}[X]$ is not a field, where $g$ is a non constant polynomial in $\mathbb{Z}[X]$?   (The key is to show that $f : \mathbb{Z}[X]/g\mathbb{Z}[X] \to \mathbb{Z}/p\mathbb{Z}$ is not injective.)  edit 1: Now I'm trying to find the characteristic of the given ring. Consider the map: $g: \mathbb{Z} \to \mathbb{Z}[X]/g\mathbb{Z}[X]$, where $m \mapsto 1 + 1 + \cdots + 1 $ (m times). Since the $\ker f$ is an ideal of $\mathbb{Z}$, it has to be of the form $n\mathbb{Z}$, where $n$ is the characteristic. So I guess the characteristic is $0$...no?  edit 2: Thank you for all the replies below! I looked through the answer but it's a little advanced for me... I figured it out eventually using some more basic ideas and I'll put it here when I get some time...Thanks again!  edit 3:  My solution:  Step 1: Prove there is a $a \in \mathbb{Z}$ such that $g(a) \neq 0, \pm 1$ and let $p$ be a prime number that divides $g(a)$.  Suppose $g(X) = \sum_{i = 1}^na_iX^i$, then $g(X) = 0$ has at most $n$ integer solutions, $p(X) = g(X) -1 = 0$ has at most $n$ integer solutions, and so is $q(X) = g(X)+1 = 0$. There are at most $3n$ integers that satisfy $g(a) = 0,\pm1$, but $\mathbb{Z}$ is infinite, so there has to be a $a \in \mathbb{Z}$ such that $g(a) \neq 0, \pm 1$.  Step 2: Prove that there is a unique well-defined surjective morphism of rings $f: \mathbb{Z}[X] \to \mathbb{Z}/p\mathbb{Z}$ sending $X$ to $a$ mod $p$ and that it yields a homomorphism of rings $\phi: \mathbb{Z}[X]/g\mathbb{Z}[X] \to \mathbb{Z}/p\mathbb{Z}$.  Observe that $g\mathbb{Z}[X]$ is in the kernel, so the map factor through $\mathbb{Z}[X]/g\mathbb{Z}[X]$.  Step 3: Show that the resulting map is not injective.  Note that the map $\phi$ is injective if and only if $\ker f = g\mathbb{Z}[X]$, but actually $g\mathbb{Z}[X] \subset \ker f$. Alternatively, according to my prof, if the map was injective, $\mathbb{Z}[X]/g\mathbb{Z}[X]$ would identify with a subring of $\mathbb{Z}/p\mathbb{Z}$ so it would have characteristic $p$, contradiction!  Step 4: We use that if $g: K \to A$ where $K$ is a field, then $g$ is either injective or the zero map. Since the above map is not a zero map and we proved that it's not injective, $\mathbb{Z}[X]/g\mathbb{Z}[X]$ can't be a field.
Let $f$ be a nonconstant element of ring $\mathbb Z[x]$. Prove that $\langle f \rangle$ is not maximal in $\mathbb Z[x]$.   Let us assume $\langle f \rangle$ is maximal. Then $\mathbb Z[x] / \langle f \rangle$ would be a field. Let $a \in \mathbb{Z}$. Then $a + \langle f \rangle$ is a nonzero element of this field, hence a unit. Let $g + \langle f \rangle$ be its inverse. Then $a g - 1 \in \langle f \rangle$, hence $ag(x)-1 = f(x)h(x)$ for some $h \in Z[x]$, hence $ag(0) + f(0)h(0) = 1$, thus $(a,f(0))=1$ for all $a \in \Bbb Z$, contradiction, hence the proof.  Is my argument correct? Is there any other method?
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It seems like something I am not aware of is being inserted into the generated LaTeX.  Also is there a way for me to see the actual LaTeX that is generated and processed so that I can debug issues like this better?  Is there a way to tell the lualatex to log the actual LaTeX it tries to process?  Any other suggestions for how to debug a problem like this? We have two PHP5 objects and would like to merge the content of one into the second. There are no notion of subclasses between them so the solutions described in the following topic cannot apply.    //We have this: $objectA-&gt;a; $objectA-&gt;b; $objectB-&gt;c; $objectB-&gt;d;  //We want the easiest way to get: $objectC-&gt;a; $objectC-&gt;b; $objectC-&gt;c; $objectC-&gt;d;   Remarks:   These are objects, not classes. The objects contain quite a lot of fields so a foreach would be quite slow. So far we consider transforming objects A and B into arrays then merging them using array_merge() before re-transforming into an object but we can't say we are proud if this. Using math in TikZ How do I press Ctrl+Alt+Delete in VirtualBox? I'm running VirtualBox with Windows Vista as host OS and Windows Server 2008 as guest OS. To login to Windows Server I have to press Ctrl+Alt+Delete, but my host OS reacts on that so I can not send it to VirtualBox. How can I do it? Variable is already defined in method lambda How do I increase the render distance? I have created a simple model here   When rendering it the end of it is cut off   How can I get the camera to render a longer distance? Is there a way to switch the mesh selection mode by a shortcut? Is there a shortcut for switching between these three modes? How to plot a surface from a set of data? Disclaimer: I already read questions like  and  but these did not solve my problem.  I have this set of data:  x       y       z 2.17    0.001   0.82044815 2.17    0.002   0.82345825 2.17    0.004   0.82679255 2.17    0.008   0.83334715 2.17    0.016   0.84395915 2.17    0.032   0.8584953 2.21    0.001   0.77582165 2.21    0.003   0.78520505 2.21    0.009   0.80205985 2.21    0.027   0.83085105 2.24    0.001   0.7227885 2.24    0.002   0.73391615 2.24    0.005   0.7543979 2.24    0.015   0.78798745 2.24    0.003   0.74176635 2.24    0.009   0.77064805 2.24    0.027   0.81042375 2.26    0.001   0.66545585 2.26    0.003   0.7012046 2.26    0.005   0.721067 2.26    0.009   0.7447984 2.26    0.015   0.76715245 2.26    0.027   0.794177 2.27    0.001   0.62916195 2.27    0.003   0.6774642 2.27    0.009   0.72961785 2.27    0.027   0.7861086 2.28    0.001   0.5750828 2.28    0.003   0.65059675 2.28    0.005   0.6802631 2.28    0.009   0.7145367 2.28    0.015   0.74447695 2.28    0.027   0.7774403 2.29    0.001   0.51357255 2.29    0.002   0.581053 2.29    0.003   0.6173075 2.29    0.009   0.6972096 2.29    0.027   0.76793225 2.31    0.001   0.36997965 2.31    0.002   0.474415 2.31    0.003   0.53649295 2.31    0.009   0.6587164 2.31    0.016   0.70870255 2.31    0.027   0.7482423 2.31    0.05    0.7912395 2.34    0.001   0.2204104 2.34    0.002   0.316308 2.34    0.003   0.39256745 2.34    0.004   0.45240835 2.34    0.009   0.5883453 2.34    0.016   0.6590771 2.34    0.027   0.71444205 2.34    0.05    0.7690014 2.38    0.001   0.13286995 2.38    0.002   0.1828288 2.38    0.004   0.2980268 2.38    0.008   0.4507145 2.38    0.016   0.58417075 2.38    0.032   0.6833616   And if I plot it in Mathematica (with ListPlot3D[data]) it looks, without further tweaking and fiddling around, like:    But if I try this pgfplots code  \begin{tikzpicture}   \begin{axis}     \addplot3 [surf] table {data.txt};   \end{axis} \end{tikzpicture}   I only get this:    What do I have to do to get a surface like in Mathematica (color, smoothness, style, etc. does not matter, just a closed surface)? Why should edits be at least six non-space characters? Earlier today I got this message:  Oops! Your edit couldn't be submitted because:   Edits must be at least 6 non-space characters; is there something else to improve in this post?   However, the purpose of my edit was to change the formatting for a message to show code.  There may have been other grammatical mistakes that could have been, but in general it is possible that there would not be.  What's the correct protocol to handle a situation like this? Proving $\arg(zw)=\arg(z)+\arg(w)$ This is my attempt I know this is incomplete or may even be wrong.   Let $θ_1 \in \arg(z)$ and $θ_2 \in \arg(w)$. Then, $θ_1+θ_2 \in \arg(z)+\arg(w)$. Also, $θ_1+θ_2 \in \arg(zw)$. Is this sufficient for the proof or correct at all? Hope someone could help me out. Thanks.   EDIT: Just to avoid any confusions I will add how $\arg z$ is defined.   $\arg z=\{\text{Arg}\, z+2\pi k\mid k\text{ is an integer}\}$ Class imbalance in Supervised Machine Learning