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Is there a standard way to understand contructions like $(?):=F\left[X\right]/\left\langle a_{n}X^{n}+\cdots+a_{1}X+a_{0}\right\rangle $, where $F$ is a field ? Because I constantly have to do exercises with contructions like these and I'm really tired of having to

a) try "manually'' to figure out how the set $(?)$ looks like

b) think every time of a homomorphism such that I may apply the homomorphism theorem for rings to get an isomorphism between the above contstruction

c) think of some other clever way which tells me what $(?)$ looks like

I'm thinking of some algorithm/grand theorem, which only given the $F,a_{n},\ldots,a_{1}$ tells me how the set $(?)$ looks like.

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    It's a particular case of the quotient of a ring by an ideal. IMHO you should try to understand this general notion.2012-01-08
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    What Pierre-Yves wrote. Plus a question: how do you define $\mathbb Z/n\mathbb Z$ and do you find difficult to *figure out* how it *looks like*?2012-01-08
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    [Related question](http://math.stackexchange.com/q/97341/660).2012-01-08
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    @DidierPiau, Pierre-YvesGaillard $\mathbb(Z) / n \mathbb(Z)$ is easy, since there we have a standard way of selecting a system of representatives, with which I can then operate. In my above example I don't know of a standard way to pick such a system, hence the question - and I'm not interested in the general case, if a quotient of a ring by an ideal, because it is too general; I'm just trying to understand this example of polynomials. Maybe I didn't formulate my question correctly: Of course I know that the elements of the quotient are of the form $t+, t \in F$ (...)2012-01-08
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    (...), but what I'm interested in is a canonical way to obtain a system of representatives - which seems difficult, depending on what the polynomial is...2012-01-08
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    @Pierre-YvesGaillard was probably not pinged. // Assuming $a_n\ne0$, you could keep only the (residue classes of) the elements $X^i$ for $0\leqslant i\leqslant n-1$.2012-01-08
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    @Did Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-13
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    @JulianKuelshammer Thanks for your input. What about closing the question instead?2013-06-14

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