# Questions tagged [polynomial-rings]

This tag is used for questions on polynomials rings in an arbitrary number of variables related to different topics studied in ring theory and commutative algebra. Questions related to high-school polynomials level or similar should use the tag "polynomials".

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### Is the cardinality of the polynomial quotient ring $\mathbb{Z}_n [x] /f(x)$ always finite?

For some polynomial $f(x)$ does the polynomial quotient ring $\mathbb{Z}_n [x] / f(x)$ always have finite cardinality? Also, I have noticed that the examples of polynomial (over $\mathbb{Z}_n$) ...
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### Polynomial differentiation map

I’m studying rings of polynomials And I don’t understand the following exercise. “Let F be a field and char(F)=0 Let D : F[x] -> F[x] defined by D(f(x))=f’(x) Find image of F[x] under D” The ...
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### How to show each element of $\frac{Q[x]}{I}$ is of the form $a_0+a_1t+a_2t^2$

Consider the polynomial ring $Q[x]$. Let $p(x) = x^3-2$. Let $I$ be an ideal generated by $p(x)$. Show that each element of $\frac{Q[x]}{I}$ is of the form $a_0+a_1t+a_2t^2$ with $a_0, a_1, a_2$in $Q$ ...
### Find the sum and product of $f(x)=3x-5, g(x)=2x^2-4x+3$ in $Z_8$.
Find the sum and product of $f(x)=3x-5, g(x)=2x^2-4x+3$ in $Z_8$. Does using high school style of solving this yields a different answer? What would be the answer?
### Show that $R[x_1,\dots x_n]/IR[x_1,\dots,x_n]\cong(R/I)[x_1,\dots,x_n]$
To simplify the notation let $X:=x_1,\dots,x_n$. Let $R$ be a commutative ring with a unit and let $I\subseteq R$ be an ideal. Let $IR[X]$ be the ideal in $R[X]$ generated by $I$. Build an ...