I want to show that $\Bbb Z[i]/(2+3i)$ is a finite field and count it's elements.  I don't really know how to show that this field is finite.  I start by trying to understand that definition of this ring, so I write   $\Bbb Z[i]/(2+3i)=${$a+bi+2+3i:a,b \in \Bbb Z$}  I don't really know what to do from here. The knowledge needed to solve this shouldn't be much farther than the Chinese Remainder theorem, so I think it should be relatively simple but I don't see how to approach it.  Any help is appreciated, thanks
Show that $\mathbb{Z}[i]/(2+3i)$ is a finite field.    By applying some isomorphisms and realizing $\mathbb{Z}[i]$ as the quotient ring of $\mathbb{Z}[x]/(x^{2}+1)$, I found that:  $\mathbb{Z}[i]/(2+3i) \cong \mathbb{Z}[x]/(x^2+1,2+3x)$  However, I run into trouble as I can't divide $x^{2}+1$ by $2+3x$ in $\mathbb{Z}[x]$. What would be another way to approach the problem?  Thanks for the help.
I need to find the closure of these sets on the ordered square: $$A = \{(1/n)\times 0 | n\in \mathbb{Z}_+\}$$ $$B = \{(1-1/n)\times \frac{1}{2}| n\in \mathbb{Z}_+\}$$ $$C = \{x\times 0 | 0&lt;x&lt;1\}$$ $$D = \{x\times \frac{1}{2}| 0&lt;x&lt;1\}$$ $$E = \{\frac{1}{2}\times y | 0&lt;y&lt;1\}$$ I know that the closure is just the set itself united with its limit points. For $A$, the set of limit points, I imagine, are $\{&lt;1,0&gt;\}$, that is, the $x$ coordinate is $1$ and the $y$ is $0$. But  is says the reverse. What am I getting wrong? Following the same reasoning, $B' = \{&lt;1,\frac{1}{2}&gt;\}$ $C' = \{&lt;0,0&gt;,&lt;0,1&gt;\}$ What am I doing wrong? How do you empty the buffers and cache on a Linux system? Compute $e^{e^z}$, where z is a complex number I am self-studying the MAT 67 linear algebra course and I got stuck at a problem in Chapter 2:     Compute the real and imaginary parts of $e^{e^z}$ for $z$ $\in$ $\Bbb C$.   I tried to expand it using Euler's formula but it seemed endless. Any help? Stupidly, I had been using a condition like this as part of a script:  if [ $(ls FOO* 2&gt; /dev/null) ] # if files named "FOO*" were downloaded  then     echo "Files found"      # ... process and email results else     echo "Not found"     # ... email warning that no files were found (against expectations) fi   That works for zero and one files named FOO*, but fails if there are more than one. From logs I found several different error messages stemming from this:  [: FOO_20131107_082920: unary operator expected [: FOO_20131108_070203: binary operator expected [: too many arguments   My question is: what is the correct way to check, in a Bash if condition, whether one or more files whose name begins with FOO exist?  GNU bash, version 4.2.25(1)-release (x86_64-pc-linux-gnu) When using the Stack Exchange App for Android, I encountered a problem when uploading multiple images to the same post.  After uploading a single image, the image Markdown is inserted normally, e.g.:  ![enter image description here][1]     [1]: http://i.stack.imgur.com/Cbh1m.jpg   As I upload more images, the list of URLs at the bottom of the post should be appended to. After uploading the above in a new post, I tap to place the cursor at the end of the first line, enter a newline and the toolbar reappears. I then upload a second image with this result:  ![enter image description here][1] ![enter image description here][2]     [1]: http://i.stack.imgur.com/Cbh1m.jpg   [2]: http://i.stack.imgur.com/V5nXV.jpg   Now I edit the image descriptions, to make sure I can keep track of all my images, place the cursor at the end of the second line and enter a newline. The edit box contains:  ![a black circle of utter blackness][1] ![the best planet earth clip art ever][2]      [1]: http://i.stack.imgur.com/Cbh1m.jpg   [2]: http://i.stack.imgur.com/V5nXV.jpg   I then upload a third image. What's interesting is that as the screen reloads, for an instant I see the best planet earth clip art ever replaced by more image markup, so the second line is this:  ![![enter image description here][2]][2]   But only for a moment. Then the screen updates, and the edit window is left in this state:  ![the best planet earth clip art ever][2] ![the best planet earth clip art ever][2] ![enter image description here][3]     [1]: http://i.stack.imgur.com/Cbh1m.jpg   [2]: http://i.stack.imgur.com/V5nXV.jpg   [3]: http://i.stack.imgur.com/6Ra5w.jpg   I correct the first line and insert a line break after the third line in preparation to upload the first image again (I really love blackness). Before uploading, the edit window looks like this:  ![a black circle of utter blackness][1] ![the best planet earth clip art ever][2] ![a fast computer][3]      [1]: http://i.stack.imgur.com/Cbh1m.jpg   [2]: http://i.stack.imgur.com/V5nXV.jpg   [3]: http://i.stack.imgur.com/6Ra5w.jpg   After uploading:  ![a fast computer][3] ![the best planet earth clip art ever][2] ![a fast computer][3] ![enter image description here][4]      [1]: http://i.stack.imgur.com/Cbh1m.jpg   [2]: http://i.stack.imgur.com/V5nXV.jpg   [3]: http://i.stack.imgur.com/6Ra5w.jpg   [4]: http://i.stack.imgur.com/ILUCC.jpg   Now, instead of adding a newline, I position my cursor at the end of the third line. The toolbar isn't visible so I insert a space and then delete it. I upload another copy of the third image from this position adjacent to other image markdown, and now the result is:  ![a fast computer][3] ![the best planet earth clip art ever][2] ![a fast computer][3]![enter image description here][4] ![enter image description here][4]      [1]: http://i.stack.imgur.com/6Ra5w.jpg   [2]: http://i.stack.imgur.com/V5nXV.jpg   [3]: http://i.stack.imgur.com/6Ra5w.jpg   [4]: http://i.stack.imgur.com/jG0fk.jpg   [5]: http://i.stack.imgur.com/ILUCC.jpg   Notice:   The new image, the fifth I have uploaded, goes in as image [4]. Not a problem, so far, but: The previous image description is still linked to image [4], now a different URL; and The first URL in the list has been overwritten by the URL of the third image, which happens to be immediately adjacent on the line where I inserted the most recently-uploaded image.   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Most legendary creatures get three Legendary Resistances per day, which they can use to automatically succeed on a saving throw.  A Concentration check is technically a Constitution saving throw, so does that mean that a creature can use a Legendary Resistance to automatically pass a Concentration check, or is it not considered a saving throw? Create element in pdf that doesn't print to paper Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$ I need to show that  $$ \int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3} $$  I have already derived the result $\int_0^\infty \frac{\sin^2x}{x^2} = \frac{\pi}{2}$ using complex analysis, a result which I am supposed to start from.  Using a change of variable $ x \mapsto 2x $ :  $$ \int_0^\infty \frac{\sin^2(2x)}{x^2}dx = \pi $$  Now using the identity $\sin^2(2x) = 4\sin^2x - 4\sin^4x $, we obtain  $$ \int_0^\infty \frac{\sin^2x - \sin^4x}{x^2}dx = \frac{\pi}{4} $$ $$ \frac{\pi}{2} - \int_0^\infty \frac{\sin^4x}{x^2}dx = \frac{\pi}{4} $$ $$ \int_0^\infty \frac{\sin^4x}{x^2}dx = \frac{\pi}{4} $$  But I am now at a loss as to how to make $x^4$ appear at the denominator.  Any ideas appreciated.  Important: I must start from $ \int_0^\infty \frac{\sin^2x}{x^2}dx $, and use the change of variable and identity mentioned above Warning or confirmation on new tag creation In retrospect of this , I would like to ask the community how they feel about a warning or confirmation (or just special highlighting) on new tag creation.  Everybody hates stock "are you sure?" confirmation, but a lot of tags get created by mistake (and then used ad nauseam).   For example, I think I might have created the  tag by accident. Assuming the right tag was , there would be two tags for c# 4.0, until they are made synonyms. With a warning, some accidental tags could be prevented. How do I determine how many spell slots I have when multiclassing? How do I determine how many spell slots I have if I multiclass into more than one spellcasting class? For example, I have a character that is a level 3 Wizard, a level 5 Eldritch Knight Fighter, a level 4 Arcane Trickster Rogue, a level 4 Paladin, and a 4 Warlock. How many spell slots do I have and how do I determine this? Java: GUI - JButton opening new JPanel and reading the JTextFields Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits. Markdown help page is broken (badly presented/formatted) on every site In the  I see:    Is that what the help is supposed to say and look like? Because it doesn't make sense to me.  As animuson points out in a comment, all of the highlighted example text is being shifted to the very left of each line. The entire page is broken. App Store is stuck in a loop trying to download a free app -- how do I make it stop? My App Store keeps saying this item is free in the store, that this item will be added to your purchases and it then says cancel or download.   I click download and it won't download and keeps popping up the same message over and over again.  How do I make it stop? I can't download any apps. Privilege banner buggy? Or am I just confused? On the main Stack Overflow site, I arrived and had a banner: "Congrats, you've gained the privilege – create wiki posts and 3 other privileges" This confuses me, because I only see two privileges listed at 100%. Is the banner just buggy?  Or is there a better place to see my privileges that the page linked to by that banner? So basically, I'd like to use iwconfig to connect to my wifi network when I'm not inside X. 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