In Chapter 4 of Lang's Algebra, Exercise 8 states: "Let $A$ be a commutative entire ring and $x$ a variable over $A$. Let $a,b \in A$ and assume that $a$ is a unit in $A$. Show that the map $x \mapsto ax+b$ extends to a unique automorphism of $A[x]$ inducing the identity on $A$. What is the inverse automorphism?" I showed that the inverse is $a^{-1}(x-b)$. Then Exercise 9 states: "Show that every automorphism of $A[x]$ is of the type described in Exercise 8."  It is clear to me that we must have $x \mapsto ax+b$ for some $a,b$, but why must an automorphism of $A[x]$ induce the identity on $A$?  (I explained why this is not a duplicate of the problem referenced in the comments.)
Suppose that $A$ is an integral domain. If $a,b$ are element of $A$ and $a$ is a unit, then how come the change of variable $X\mapsto aX+b$ extends to a unique automorphism of the polynomial ring $A[X]$, which equals the identity map when restricted to $A$?  Thanks for any ideas or ways to get started.
How to convert a one digit number to a two digit number I have a table that represents a timetable for bus, so each cell contains time of departure as hour followed by minute, both as two digits, but sometimes I don't have two digits,just one so I have to switch eg. from 12 4 -&gt; 12 04. How is it possible? How many players can play split-screen, and what game modes are (or aren't) available?   How many players can play local co-op? How many players can play local multiplayer? How many players can play online multiplayer from a single console? What is a tensor? I have a pretty good knowledge of physics, but couldn't deeply understand what a  is and why it is so fundamental. Show $\sin x = bx$ has at least 3 real solutions. Show $\sin x=bx$ has at least 3 real solutions for every $b$, where $-\dfrac{2}{3 \pi} &lt; b &lt; \dfrac{2}{\pi}$.   Some useful facts I am trying to use:  Intermediate value theorem: If $f$ is a continuous real-valued function on $[a,b]$ and $d \in \mathbb{R}$ satisfies $f(a)&lt; d&lt; f(b)$, then there exists a point $c$ in $(a,b)$ such that $f(c) = d$.   Corollary: If $f$ is a continuous real-valued function on $[a,b]$, then $f([a,b])$ is a closed interval. I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work? Wikipedia defines the notion of an abelian category as follows ().     A category is abelian iff         it has a zero object,   it has all binary products and binary coproducts, and   it has all kernels and cokernels.   all monomorphisms and epimorphisms are normal.      It later explains that an abelian category can naturally be enriched in $\mathbf{Ab},$ as a result of the first three axioms above.  How does this actually work? I am a physics Master's student.  I have good grades, with CGPA above 9(/10)(However, all courses in my university are graded on a curve so the actual marks are significantly lower than 90%). This has been my marks throughout my undergraduate and graduate courses and I was happy because this was above average. Also, my summer projects in UG were usually computational and any difficulty I had in theoretical aspects I attributed it to my inexperience in advanced Physics.   However, recently I have started to work on pure theoretical project and while reading for the project I realized that I consistently miss some details from each topic. I understand the general idea, can re-derive most of the equations(with some effort) however when I discuss things with my professor or when I read further topics I realize that I have missed certain key-details. This is exactly similar to the case of me missing a few key questions in my exams and getting that 9 grade point instead of 10(or at-least I think so). I was not worried about this before but now I realize that there is some flaw in the way I study (but I do not know what).   Since I want to apply for Physics PhD I realized that I will need to have solid concepts and it would be best to correct my mistakes but I do not know how or even what could be the reason for my imperfect understanding.  Any direct suggestion would help or even certain general guidelines on what I can do.    Things I've tried recently -   1) I've started to written down almost everything I read. This is slightly laborious but is certainly helping. It also helps me to gloss over the things I read previously when I come back the next day.  2) I completely avoid smartphone when I'm library/study room so that I won't get distracted. Not sure if it helps me solve the above problem but certainly helps me study better in general.  3) I've tried to read the topics from several sources. The books I am reading is somewhat like a collection of research papers and hence does not have a exercises. So I though reading from various sources might help get better ideas but it's not really helping because most other sources are too high level as an introductory text. But this will probably help me in cases where it's practical to do this.  4) Somewhat similar to point (1)- I am trying reread, whenever I can, what I've done until that point. But not sure how much this will help me in filling the gaps. How To Make A Solid Object Drop To The Ground Like It Was Liquid Given $A$ and $B$ positive-definite matrices and $Q$ unitary matrix, prove that if $A = BQ$, then $A=B$. $Q$ is unitary, so $QQ^*=I$ If $A$ and $B$ are positive-definite, than $A=A^*$ and $B=B^*$. $A^*=(BQ)^*=Q^*B^*$ $A^2=AA=AA^*=(BQ)(Q^*B^*)=B(QQ^*)B^*=BIB^*=B^2$ $A^2=B^2$. I don't know how to use the fact that $A$ and $B$ are positive-definite to finish the proof. When I visit a website it sometimes puts cookies on my browser. Usually cookies keep track of browsing info from that particular site. If I don't delete those cookies and browse other websites, will the cookies be able to know which site I am browsing? Also, can any website read browsing history which is saved by browsers? Can any website know about my browsing history by help of cookies, browsing histories saved by browsers, or by any scripting language e.g. javascript? How is the passage of time handled in Pokemon? How to switch render source (VSE or Camera)? If you create a scene for the Video Sequence Editor and start working on a project Blender will render the output from the VSE. But if you want to later switch it to render your 3D scene again (the camera output) where is the setting to select the source?  Usually I keep a separate .blend for VSE edits so I don't need to switch, but I feel like I ought to learn how to do this as it's pretty basic. I would expect to find this in the Render context but I see no place to specify the source.  Thanks. Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$ I am wondering whether there exists a function such that:  $$\lim_{x \rightarrow a}f(x)=\infty$$  at some point $a$ on the real axis but yet,  $$\int_{-\infty}^{+\infty}\left|f(x)\right|\ dx&lt;\infty$$  Does the fact that a function is unbounded imply that it has no finite integral? Possible Duplicate:          As a English non-native speaker it is difficult for me to understand when I must use present perfect or past simple because in my official language there isn't the present perfect tense. I know that present perfect is related to something that happened in the past but its result is important now. But for instance:        The message has been sent.   The message was sent.      Please help me point me out any examples/contexts where I should use 1. instead of 2. I have asked this question before  The first guy solved it in a way, but  i would like to know is there any efficient way to do so, because i did the same for N = 13 and N = 15  which took long time.  This problem is from a pen and paper test, ZIO-2018 Can i solve it with an efficient algorithm, well i am pretty beginner in algos, so if there is one kindly descibe it how can i do so?  Thank you Cannot remove any bluetooth device How to create a fancy table of contents for Beamer How can I create a fancy-looking table of contents for a Beamer presentation?    \documentclass{beamer} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc}  \begin{document}  \frame{ \tableofcontents }  \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{}  \end{document} I am not a administrator and I would like to find the administrator password. I would not like to reset it however. Is there anyway to find and not reset the administrator password? Travel to Ireland (Dublin) with U.K. Visitor visa Arch on WSL: libQt5Core.so.5 not found, despite being installed I was hit by some hardcore power-tripping forum mods on the Arch Linux forum, so here I am.  I'm trying to run GUI apps on WSL, such as dolphin or kate. However, none of them can find libQt5Core.so.5, despite it being installed in /usr/lib as you'd expect. Here's a terminal transcript:  /u/lib » DISPLAY=:0.0 dolphin dolphin: error while loading shared libraries: libQt5Core.so.5: cannot open shared object file: No such file or directory /u/lib » pwd /usr/lib /u/lib » ls -la | grep Qt5Core -rw-r--r-- 1 root root     1143 Aug  7 06:05 libQt5Core.prl lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so -&gt; libQt5Core.so.5.11.1* lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so.5 -&gt; libQt5Core.so.5.11.1* lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so.5.11 -&gt; libQt5Core.so.5.11.1* -rwxr-xr-x 1 root root  5234312 Aug  7 06:05 libQt5Core.so.5.11.1* /u/lib » ldd `which dolphin`         linux-vdso.so.1 (0x00007fffdb426000) ...         libQt5Core.so.5 =&gt; not found /u/lib » pacman -Qk qt5-base qt5-base: 3546 total files, 0 missing files   Note that other GUI apps work fine - anything compiled against GTK, including Firefox, work perfectly. I just have to set the $DISPLAY variable. Not setting the var doesn't help, as you'd expect.  Any ideas? Thanks!