I'm a little bit confused by the notation &quot;$K[t]$&quot;, what does it denote? I know that $K[X]$ is the polynomial ring over $K$, so the elements $P$ of $K[X]$ look like $P(X) = \sum_{n=0}^\infty k_n X^n$ with $k_n \in K$. Is $K[t]$ then just the image of the homomorphism $P \mapsto P(t)$ with $P \in K[X]$? Wouldn't that just be same as $K(t)$, which is the smallest field that contains both $K$ and $t$? What's the difference?
I have a question about notation specifically square brackets $[$ and round brackets $($.  My textbook doesn't explain any of this and I cannot find a reliable source online to confirm the difference.  So my question is: What is the difference between round brackets and square brackets in terms of notation in Field Theory? For example, I see $F(x)$ and $F[x]$ in my textbook and I've always assumed they were the same thing. But apparently they're not. Is there ever a time they're the same?  I wanted to know the difference, it may be a silly question but it's something I want to make sure I understand.
Possible Duplicate:          Is there some rule governing the following, or similar, subject-auxiliary inversions (*"Rarely they do see the light of day", *"Never I have been so insulted")?     In fancy sit-down restaurants, you can order a large meal and halfway through the main course, take a little dead cockroach or a piece of glass out of your pocket and place it deftly on the plate. Jump up astonished and summon the headwaiter. "Never have I been so insulted. I could have been poisoned" you scream slapping down the napkin. You can refuse to pay and leave, or let the waiter talk you into having a brand new meal on the house for this terrible inconvenience. ()      Rarely do they see the light of day; it's generally judged to be in no one's commercial interest that they should. Just occasionally, however, like a seemingly dormant volcano, they explode into the open. () Samsung galaxy S4 Blacked out Screen need to retreive files What the #$@&%*! is that called? Find the value of $a^2-b^2+c^2$ Clerical error in UK visa application, but form already submitted - what should I do? I'm from the Philippines, and I plan on travelling to the UK with my sister to visit a relative.  I have just submitted my visa application and have already paid online, when I noticed that I've made a clerical error. In the 'visit or stay' portion of section "Do you have any friends or relatives in the UK?", I clicked the 'no' box, instead of 'yes'.  My sister's application is correct. She selected 'yes'. Our appointment is due next month. I am planning to email them about this, and also inform them on the day of application submission.  Is this the right path to take? What do they do with all the plates and cups in Star Trek? Whenever someone in Star Trek creates food or drink in the replicator it is generated in the bowl, plate, or cup. It seems like after a few months they would have piles of used dishes from replicator food while making more for every meal. 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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). 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Makeindex error when ! used in \index in align environment See MWE below. The symbol ! in the \index causes an error. This is only the case in align environments (not in equation). So it seems to have to do something with amsmath. If ! is put in an \mbox there is no error. But then of course the font size is not adjusted to subindices. Curiously enough, \index{$C_{K!!}$} (two exclamation marks) doesn't cause any error.    \documentclass{article} \usepackage{amsmath} \usepackage{imakeidx} \makeindex \begin{document} \begin{align} y\index{$C_{K!}$} \end{align} \printindex \end{document}     Any solutions suggested?