Let $F$ be a finite extension of the finite field $K$ and $\alpha \in F$. The mapping $L: \beta \in F \mapsto \alpha \beta \in F$ is a linear transformation of $F$, considered as a vector space over $K$. Prove that the characteristic polynomial $g(x)$ of alpha over $K$ is equal to the characteristic polynomial of the linear transformation $T$; that is $g(x) = det(xI - L)$, where $I$ is the identity transform.
This is problem 31b) from Dummit and Foote chapter 14.2. I am looking for a hint on how to attack the problem, since I have been thinking about it for a couple of hours but I don't even know where to start. The problem says:  Let $K$ be a finite extension of $F$ of degree $n$. Let $\alpha$ be an element of $K$. Prove that the minimal polynomial for $\alpha$ over $F$ is the same as the minimal polynomial for the linear transformation $T_{\alpha}$. In this problem, $T_{\alpha}$ is an $F$-linear transformation of $K$ that arises from $\alpha$ acting by left multiplication on $K$.  I appreciate any helpful suggestions on how to start attacking the problem or any possible hint. Thanks!
Can I sync or transfer save games and settings from one platform to another? I would like to know if the same Minecraft account used on a computer can be used on a different platform like the Xbox 360? I am not asking about playing on each simultaneously as each platform has been purchased - I am asking if you can cross play on multiple platforms with the same account. Basically I would like to know if all the saved stuff and settings from one platform will carry over to another. PC started having screen glitches after update I'm not sure what's caused it. I haven't changed any system settings. The only thing I've done that may have broken something is updating my software. My update was: upgrade linux-generic-hwe-20.04:amd64 5.8.0.48.54~20.04.32 5.8.0.49.55~20.04.33 upgrade linux-image-generic-hwe-20.04:amd64 5.8.0.48.54~20.04.32  upgrade linux-headers-generic-hwe-20.04:amd64 5.8.0.48.54~20.04.32  upgrade linux-image-extra-virtual:amd64 5.4.0.70.73 5.4.0.71.74 upgrade linux-image-generic:amd64 5.4.0.70.73 5.4.0.71.74 upgrade linux-libc-dev:amd64 5.4.0-70.78 5.4.0-71.79 upgrade libseccomp2:amd64 2.4.3-1ubuntu3.20.04.3 2.5.1-1ubuntu1~20.04.1 upgrade nomachine:amd64 7.3.2-3 7.4.1-3```  Removing nomachine did not solve the problem. It was far worse before, I'm not sure how to describe exactly, but my entire screen's UI was glitching out. So I rebooted and that stopped, but now the terminal text looks like this:  I haven't changed any terminal settings. This seems to be occurring in the entire OS. Here is another example:  Not sure how to go about troubleshooting this? I don't have the time to reinstall the OS, nor do I think I should have to because this is just ridiculous and shouldn't happen. I'm not aware of anything I've done to make the system unstable. Edit Earlier I also tried booting the PC. It failed and got stuck on some command in a terminal (didn't manage to take a photo of what it said). So I rebooted into recovery mode with both available kernels. I was getting strange glitches like this:  That didn't work. So I rebooted the PC normally, and now I somehow logged in fine. Now I'm getting popups saying to report a system error. It says this:  Environment  Ubuntu 20.04.2 LTS GNOME 3.36.8  sudo lshw -class display   *-display        description: VGA compatible controller        product: Xeon E3-1200 v3/4th Gen Core Processor Integrated Graphics Controller        vendor: Intel Corporation        physical id: 2        bus info: pci@0000:00:02.0        version: 06        width: 64 bits        clock: 33MHz        capabilities: msi pm vga_controller bus_master cap_list rom        configuration: driver=i915 latency=0        resources: irq:33 memory:f7800000-f7bfffff memory:e0000000-efffffff ioport:f000(size=64) memory:c0000-dffff  I also did , which passed with 0 errors. Let me know if more details are required. Why do the Localization of a Ring This question may be a bit vague, but neverthless, i would like to see an answer. Wikipedia tells me that:   In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring.   It is clear that for integral domains, we have the Field of Fractions, and we work on it. It's obvious that a Field has multiplicative inverses. Now if we consider any arbitrary ring $R$, by the definition of localization, it means that we are adding multiplicative inverses to $R$ thereby wanting $R$ to be a division ring or a field. My question, why is it so important to look at this concept of Localization. Or how would the theory look like if we never had the concept of Localization. Can't install Oracle Java 8 in Ubuntu 16.04 I want to extrude a curve in two directions.    Or join both curves into one, is it possible? I've tried copying it and then join segments, but I don't see Remove Doubles option while working with curves. @ variables in Ruby on Rails youtube-dl failed to extract signature Is there any way to generalise this problem? How can we move an object with zero velocity? Let $X$ be a normed vector space.   Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete if and only if $X$ is Banach.     I have found the first part in a book, but I do not how to prove part 2. Thanx regards How to render lightning in Cycles How can you render a lightning bolt using Cycles? Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $ What is a "closed" or "duplicate" question? Questions can be closed or marked as duplicates.  What does it mean for a question to be &quot;closed&quot;? What does it mean for a question to be marked as a &quot;duplicate&quot;? Who can close a question? What are the reasons for closing a question ? Is closure the end of the road for a question? When are closed questions eligible for deletion?   Related        For more information, see &quot;&quot; in the . Why is Steam only showing me "Buy" instead of "Play" option for a game shared via Family Sharing? I thought the whole point of Family Sharing was so that multiple copies of the game didn't have to be bought. I have no problem downloading the game, but instead of "Play", my only option once it's downloaded is "Buy". Am I missing something? Prettiest way to typeset "C++" (cplusplus)? Writing "C++" in plain text results in an ugly setting, as the '+' signs are too big and too spaced:    I've seen around the web several marcos for typesetting the "C++" symbol, ranging from mild kerning to shrinking, raising and lowering the '+' signs.  What is your version of a \cpp{} macro? Is there a value that can be assigned to my time, i.e. an hourly rate, so that I can be given a receipt to remit to CRA come Income Tax Filing time?  I live in Canada. Get the index of the nth occurrence of a string? I have seen similar questions on the matter (, ) of adding vertical space and not breaking vertical lines, the solutions suggested are to add space after the \\ using \\[0.1cm] for example or changing the \arraystretch which I do not want to do. Using \\[0.1cm] allows me to add space just before a \hline but not after as well.  How can I add vertical space after a \hline without breaking vertical lines in a table?  Please find my MWE below:  \documentclass{scrreprt} \usepackage{setspace} \usepackage[left=2.5cm,right=2.5cm,top=3cm,bottom=3cm]{geometry} \usepackage{array,tabularx} \doublespacing  \begin{document} \begin{table}[htb]   \centering   \onehalfspacing   \caption{Table of info.}      \begin{tabular}{c | c c}  &amp; C/S Analyser &amp; O/N/H Analyser \\  &amp; EMIA 320 V2 &amp; EMGA 830 \\[0.1cm]    \hline    \noalign{\vskip 0.1cm} Sensitivity (ppm) &amp; 0.1 &amp; 0.001 \\ Furnace type &amp; Induction &amp; Impulse \\ Carrier gas &amp; Pure O2 &amp; Pure He \\[0.1cm]    \hline     \end{tabular}    \doublespacing   \label{tab:chem} \end{table} \end{document} Why Was Peter Pettigrew Sorted To Gryffindor? Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. (using disk/washer)  I saw no example of this problem anywhere.. I saw an example how to solve it without calculus but I'm afraid that'll be deductions on my test.. I have the volume of a cylinder and the method of area between two curves but this is out of my reach.... I don't know what I can try...