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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. – Pierre-Yves Gaillard Jan 8 '12 at 10:08
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? – Did Jan 8 '12 at 10:17
Related question. – Pierre-Yves Gaillard Jan 8 '12 at 10:46
@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 $t+<a_n X^n \cdots + a_0>, t \in F$ (...) – el le Jan 8 '12 at 19:51
@Pierre-YvesGaillard was probably not pinged. // Assuming $a_n\ne0$, you could keep only the (residue classes of) the elements $X^i$ for $0\leqslant i\leqslant n-1$. – Did Jan 8 '12 at 20:11

As Did remarks:

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

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