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".

321 questions
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How to deal with polynomial quotient rings

The question is quite general and looks to explore the properties of quotient rings of the form $$\mathbb{Z}_{m}[X] / (f(x)) \quad \text{and} \quad \mathbb{R}[X]/(f(x))$$ where $m \in \mathbb{N}$ Classic examples of how one can treat such rings is…
Mathmo
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Classification of commutative rings $R$ satisfying $\dim(R[T])=\dim(R)+1$

Let $R$ be a commutative ring. Then $\dim(R[T]) \geq \dim(R)+1$. Is there a classification of those commutative rings with the property $\dim(R[T])=\dim(R)+1$? Every Noetherian commutative ring has this property, but also every $0$-dimensional…
9
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2 answers

Subrings $A$ of $\mathbb{F}_p[x]$ such that $\dim_{\mathbb{F}_p}\mathbb{F}_p[x]/A=1$.

Suppose we have the ring $\mathbb{F}_p[x]$ of polynomials with coefficients in the Galois field with $p$ prime number elements. I want to find all subrings $A$ containing the identity such that when considered as vector spaces over $\mathbb{F}_p$,…
8
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3 answers

Show $\mathbb Z[x]/(x^2-cx) \ncong \mathbb Z \times \mathbb Z$.

For integers $c \ge 2$, prove $\mathbb Z[x]/(x^2 - cx) \ncong \mathbb Z \times \mathbb Z$. (Hint: for a ring $A$, consider $A/pA$ for a suitable prime $p$.) I'm not entirely sure what the hint means, and I don't really have an idea for an…
7
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2 answers

Prime ideals in $\mathbb{Z}[X], \mathbb{Q}[X], \mathbb{F}_{19}[X]$.

Old exam question Consider the following ideals : $I = (X^{2018}+3X+15)$; $J = (X^{2018}+3X+15, X-1)$; $K = (X^{2018}+3X+15, 19)$. Determine whether they are prime ideals in $\mathbb{Z}[X], \mathbb{Q}[X], \mathbb{F}_{19}[X]$, respectively. As…
7
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3 answers

Show that the ideal generated by two polynomials is actually equivalent to the ideal generated by a single polynomial

I've been asked to show that the ideal $I = \langle x^3 - 1, x^2 - 4x + 3\rangle$ in $\mathbb{C}[x]$ is equivalent to $I = \langle p(x)\rangle$ for some polynomial $p(x)$. Now, I'm not quite sure what the ideal generated by two polynomials looks…
user3002473
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6
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Prove that $\mathbb{F}_2[X] / \left(x^3 + x + 1 \right)$ is a field with 8 elements

Just want to know if my proof is correct. First of all, it is easy to check that $f(x)=x^3+x+1$ is irreducible over $\mathbb{F}_2$. This implies that $\frac{\mathbb{F}_2[X]}{(f(x))}$ is indeed a field. In particular, it must be a field extension of…
6
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2 answers

To what ring is $\mathbb{Z}[X,Y,Z]/(X-Y, X^3-Z)$ isomorphic?

The problem: Let $(\mathbb{Z}[x,y,z],+,\cdot)$ be the ring of polynomials with coefficients in $\mathbb{Z}$ in the variables $x$, $y$ and $z$ and the obvious operations $+$ and $\cdot$. Let $(x-y, x^3-z)$ be the ideal generated by $x-y$ and $x^3-z$.…
6
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1 answer

Show the following polynomial is Irreducible over the given ring

Studying for a qualifying exam and this practice problem flat out has me stumped. I wish to show that the polynomial $(y+8)^2x^3 - x^2 + (y+7)(y+8) - y - 12$ is irreducible over $\mathbb{Q}[x, y]$. My thought was to use Eisenstein's for…
6
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2 answers

$\Bbb R[X,Y]/(F) \cong \Bbb R[Z]$ and $F_X G_Y - G_X F_Y \in \Bbb R^*$

Hope this isn't a duplicate. I was trying to solve the following problem : Let $F,G \in \Bbb R[X,Y]$ satisfy $\Bbb R[F,G]= \Bbb R[X,Y]$. Prove that : (i) $\Bbb R[X,Y]/(F) \cong \Bbb R[Z]$ , for some $Z$ and (ii) $F_X G_Y - G_X F_Y \in \Bbb R^*$…
user422112
6
votes
1 answer

Factoring $x^n + 1$.

By the Fundamental Theorem of Algebra, every polynomial of degree $n$ can be factored into a product of $n$ linear polynomials. As an example, since the polynomial $ x^5 +1$ has the five complex roots $$\tag{1} -1,\quad e^{\frac{\pi i}{5}}, \quad …
Mo Pol Bol
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6
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1 answer

Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain.

I want to prove that for a ring $R$, $R[x]$ is an integral domain if and only if $R$ is an integral domain. I have one direction of the proof ($R$ an integral domain implies $R[x]$) an integral domain, but I am having trouble proving the other…
5
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2 answers

$\mathbb{C}[x,y,z]/(x^2+y^2+z^2-1)$ is not a UFD

Wiki says that the coordinate ring $\mathbb{C}[x,y,z]/(x^2+y^2+z^2−1)$ of the complex sphere is not a unique factorization domain. I want to know why it is not a UFD. We denote $X,Y,Z$ the residue class of $x,y,z$. Obviously, we have…
5
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1 answer

Exercise III.5.10 in Grillet's Abstract Algebra

Let $R$ be commutative, with characteristic either $0$ or greater than $m$. Show that a root $r$ of $A \in R[X]$ has multiplicity $m$ if and only if $A^{(k)}(r)=0$ for all $k
Alex Ivanov
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If $I$ and $J$ are ideals and I know an algorithm to find $I : J$, can I use this to find the saturation of $I$ by $J$?

I know a pair of ideals $I$ and $J$ in a Noetherian ring and have an algoritm to find a generating set of the quotient ideal $I : J$. If I keep applying this algorithm successively finding generating sets for $I : J$ $(I : J) : J$ $((I : J) : J) :…
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