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".

# Questions tagged [polynomial-rings]

321 questions

**40**

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**2**answers

### 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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**9**

votes

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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…

Martin Brandenburg

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### 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 $\mathbb{F}_p$,…

take008

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**8**

votes

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### 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…

gravitybeatle

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### 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…

Jos van Nieuwman

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**7**

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### 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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**1**answer

### 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…

ThCastro

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### 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 $x^3-z$.…

beertje00

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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…

Travis62

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### $\Bbb R[X,Y]/(F) \cong \Bbb R[Z]$ 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**

votes

**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…

MathStudent1324

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votes

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### $\mathbb{C}[x,y,z]/(x^2+y^2+z^2-1)$ 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…

Kevin

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**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) :…

John Phonics

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