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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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    @JulianKuelshammer Thanks for your input. What about closing the question instead?2013-06-14

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As Did remarks:

Assuming $a_n \ne 0$, you could keep only the (residue classes of) the elements $X^i$ for $0 \leqslant i \leqslant n-1$.