Does a ring isomorphism preserves the structure of a field? The problem is the following one:  Let $f: A \rightarrow B$ a ring isomorphism. If A is a field, is B a field?  I think It's false, but I can't think of a counterexample, and if it's true I have no idea of how to prove it.
Isomorphisms preserving integral domains and fields Let $R \simeq S$ be isomorphic commutative rings with unity. Prove the following:  a). If $R$ is an integral domain then $S$ is an integral domain.  For this, I said that If I let $f: R \to S$ be an isomorphism and $(R, + , \circ), (S, \oplus, \ast)$, then let $x, y\in R$,  such that $x \circ y = 0_R$, because $f$ is an isomorphism, $f(x \circ y) = f(x) \ast f(y) = f(0_R)$. If I let $R$ be an integral domain then, as per the definition $R$ is commutative, has unity, and no zero divisors.   So $f(x \circ y) = f(x) \ast f(y) = f(0_R)$ where we have either $f(x \circ y) = f(0_R) \ast f(y) = f(0_R)$ or $f(x \circ y) = f(x) \ast f(0_R) = f(0_R)$.   Since $f$ is an isomomorphism, let $s = f(x), t = f(y)| s,t \in S$. Then $s \ast t = f(0_R)$. By definition, $S$ also has unity so $0_S$ exists. If we map $0_R \to 0_S$. Then either $s = 0_S$ or $t = 0_S$.   So then, since $S$ has unity, is commutative, and has a no zero divisors as we showed above, it is also an Integral domain.  I'm not sure I have this correctly.  b). If $R$ is a field then $S$ is a field. I don't really understand how to do this one. Since $S$ is an integral domain, it has a $0_S$. So I need to show that $S$ distributes $\ast$ over $\oplus$.  Any help with these is appreciated
Why $\dim U+\dim U^\perp=\dim V$? Where is the "Manipulate Centers Only" button in Blender 2.8? I have looked just about everywhere on the Internet and I can't find the answer to this question. Where the hell is the "Manipulate Centers Only" button in Blender 2.8???? It's the the double arrow icon here:    I seem to have enabled it somehow and I can't scale or rotate my object. If $A$ and $B$ are positive definite, then is $B^{-1} - A^{-1}$ positive semidefinite? Beamer, Babel and Tikz conflict Do the predictions of a Random Forest model have a prediction interval? I want to have a vector with a superscript:  \vec{o}^t   But this gives very bad results: the t is overlapping with the vector arrow. How do I fix this (and why does this give bad results in the first place) ? Why do clocks measure arc-length? Apologies in advance for the long question.  My understanding is that in GR, massive observers move along timelike curves $x^\mu(\lambda)$, and if an observer moves from point $x^\mu(\lambda_a)$ to $x^\mu(\lambda_b)$, then his clock will measure that an amount of time $t_{ba}$ given by the curve's arc length; $$   t_{ba} = \int_{\lambda_a}^{\lambda_b}d\lambda \sqrt{-g_{\mu\nu}(x(\lambda))\dot x^\mu(\lambda)\dot x^\nu(\lambda)} $$ will have elapsed where $g_{\mu\nu}$ is a metric on spacetime with signature $(-,+,+,+)$.    Why is this so?     Here is how I would attempt to justify this fact in special relativity with $g_{\mu\nu} = \eta_{\mu\nu}$.  Consider an inertial observer $O$ in $\mathbb R^{3,1}$, and suppose that this observer sees a clock, which I'll call observer $O'$ moving around on a curve $x^\mu(\lambda)$.  If $O'$ were also an inertial observer, then given any event with coordinates $x^\mu$ as measured by $O$, observer $O'$ would measure the coordinates of the event to be $x'^\mu = \Lambda^\mu_{\phantom\mu\nu} x^\nu + x_0^\mu$ for some Lorentz transformation $\Lambda$.  If $O'$ is not inertial, then this is no longer true, and there is some more complicated family of transformations, say $T_\lambda$ between events as seen by both observers.  I would argue, however, that if we were to partition the interval $[\lambda_a, \lambda_b]$ into a large number $N$ of intervals $I_1=[\lambda_a, \lambda_i], \dots, I_N=[\lambda_{N-1}, \lambda_b]$ with $\lambda_n = \lambda_a+n\epsilon_N$ and $\epsilon_N=(\lambda_b-\lambda_a)/N$, then on each interval $I_n$, $O'$ is approximately an inertial observer in the sense that $$   T_{\lambda_n} = P_n + \mathcal O(\epsilon_N), \qquad (\star) $$ for some Poincare transformation $P_n$.  Then we would note that since $O'$ is stationary in his own reference frame, he measures his worldline to have the property $\dot x'^\mu(\lambda) = (\dot t(\lambda), \mathbf 0)$ so that $$   I_{ba}=\int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x'^\mu\dot x'^\mu} = \int_{\lambda_a}^{\lambda_b} d\lambda \, \sqrt{\dot t^2} = t(\lambda_b) - t(\lambda_a) = t_{ba} $$ On the other hand the integral on the left can be written as a Riemann sum using the partition above, and we can invoke ($\star$) above to get \begin{align}   I_{ba}  &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x'^\mu(\lambda_n)\dot x'^\nu(\lambda_n)}\right] \notag\\ &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x^\mu(\lambda_n)\dot x^\nu(\lambda_n)} + \mathcal{O}(\epsilon_N^2)\right] \notag\\ &amp;= \int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x^\mu\dot x^\mu} \end{align} Combining these two computations gives the desired result.  How do others feel about this argument?  I'm not completely comfortable with it because of the assumption $(\star)$ I made on $T_\lambda$.  I imagine that in GR a similar argument could be made by invoking local flatness of the metric. Suppose $K/F$ is an extension field, and $E_1$ and $E_2$ are two intermediate subfields of finite degree. Let $E$ be the subfields of $K$ generated by $E_1$ and $E_2$. I'm trying to prove that $$ [E:F]\leq[E_1:F][E_2:F].$$  Since $E_1$ and $E_2$ are finite extensions, I suppose they have bases $\{a_1,\dots,a_n\}$ and $\{b_1,\dots,b_m\}$, respectively. If $E_1=F$ or $E_2=F$, then the inequality is actually equality, so I suppose both are proper extension fields. I think $E=F(a_1,\dots,a_n,b_1,\dots,b_m)$. Since $\{a_1,\dots,a_n,b_1,\dots,b_m\}$ is a spanning set for $E$ over $F$,  $$[E:F]\leq n+m\leq nm=[E_1:F][E_2:F]$$ since $m,n&gt;1$.  Is this sufficient? I'm weirded out since the problem did not ask to show $[E:F]\leq [E_1:F]+[E_2:F]$ which I feel will generally be a better upper bound. Prove This Inequality ${\pi \over 2} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}} \le {\pi \over 2} + 1$ $${\pi \over 2} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}} \le {\pi \over 2} + 1$$  I see I should use Riemann sum, and that $$\int_0^{\infty} {dx \over {1+x^2}} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}}$$  But how do I explain this exactly, and how to get the other half of the inequality (with ${\pi \over 2} + 1$)? This is a pretty simple question, at least it seems like it should be, about sudo permissions in Linux.  There are a lot of times when I just want to append something to /etc/hosts or a similar file but end up not being able to because both &gt; and &gt;&gt; are not allowed, even with root.  Is there someway to make this work without having to su or sudo su into root? Identify the ring $\mathbb{Z}[x]/(2x)$ Consider the quotient ring $R:=\mathbb{Z}[x]/(2x)$. Could somebody help me to identify $R$ with some well-known ring?    Thank You. How to scroll in a terminal using keyboard? How can I scroll in bash using only the keyboard? If it's not possible in bash, are there any other shells that support this? I checked the other question similar to this issue yet I could not find a proper solution for the issue I am facing for including an .svg file which is converted using inkscape with pdf+latex option set to ON, and I have also included both the .pdf and .pdf_tex file in the path. Yet I get the above mentioned error.   My code:  \begin{figure}[h!]  \centering \def\svgwidth{\columnwidth} \input{Implementation/Figs/most_changed_from_default_pos_0.pdf_tex} \end{figure}   Full error description:  !pdfTeX error: pdflatex (file /compile/Implementation/Figs/most_changed_from_default_pos_0.pdf): PDF inclusion: required page does not exist &lt;5&gt;   Any help to solve this issue would be really appreciated. I also downloaded the pdf file to check whether it has 5 pages. I could verify it, there were 5 pages in the pdf file.   The file which I want to convert is   I have added the google drive link to the file which I have used for analysis herewith How did Stannis's army get north of the wall? If Mance Rayder had to attack Castle Black to use the tunnel to get through the Wall and get out of the north, how did Stannis get his army across the Wall into the north without anyone knowing? If there was another way over the Wall wouldn't the wildings go that way too? reason for exec in wrapper scripts I have seen wrapper script examples which in a nutshell are following:  #!/bin/bash  myprog=sleep echo "This is the wrapper script, it will exec "$myprog""  exec "$myprog" "$@"   As seen above, they use exec to replace the newly created shell almost immediately with the $myprog. One could achieve the same without exec:  #!/bin/bash  myprog=sleep echo "This is the wrapper script, it will exec "$myprog""  "$myprog" "$@"   In this last example, a new bash instance is started and then $myprog is started as a child process of the bash instance.  What are the benefits of the first approach? Convert String to Integer in QGIS tables I want it to know how to convert string field values to integer values in QGIS Dufour... there are numbers and strings in the same field, when I use toreal(string) it stops and it won´t create the field. Why aren't there any common words for 'defecating' and 'urinating'? Do these two observations suffice to show that a finite boolean ring must be of the form $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$? Question: If $R$ is a finite boolean ring, then show $R \cong \mathbb Z_2 \times \mathbb Z_2 \times \cdots\times \mathbb Z_2$.   I know that   $\mathrm{char}(R) =2$  $R$ has $2^k$ elements for some $k$  (this is because, if $p\neq 2$ is a prime with $p\mid|R|$, then there exists a non-zero $x \in R$ s.t. $px=0$ and $p $ is odd $\implies x=0$, contradiction).   Does this suffice to answer my question? Spacing problem at List of Figures I want to achieve the exact same title page layout as the paper shown . For those that do not want to download the paper, the image of the title page is shown below:    The super-scripts indicating the authors doesn't have to be as shown in the example title page (e.g, a, b and c), other symbols are fine.  I am using the authblk package and know the general syntax such as:  \title{title of paper} \author[1]{author 1\thanks{details that goes into the footnote}} \author[1]{author 2\thanks{details that goes into the footnote}} \author[2]{author 3\thanks{details that goes into the footnote}} \affil[1]{university 1} \affil[2]{university 2} \renewcommand\Authands{ and } \maketitle  \begin{abstract} abstract goes here \end{abstract}   But I am not sure how to achieve the example title page shown. Any help would be appreciated!