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]
362 questions
43
votes
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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1 answer
Is the set of subrings of $\mathbb Z[X]$ countable?
Initially, I was trying to look at the subrings of $\mathbb{Z}[X]$. Since I have failed hard, I have tried to at least count them.
So I have tried to build an injection from $\{0,1\}^\mathbb{N}$ to the set of subrings of $\mathbb{Z}[X]$. Since…
Fnifni
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10
votes
1 answer
Are there uncountably many subrings of $\mathbb Z/p\mathbb Z[x]?$
For primes $p,$ are there uncountably many subrings of $\mathbb Z/p\mathbb Z[x]?$
In this question, my answer shows that the set of subrings of $\mathbb Z[x]$ is uncountable.
Indeed, we showed that if $R$ is a commutative ring (with identity) with…
Thomas Andrews
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10
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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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9
votes
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…
gravitybeatle
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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 $\mathbb{F}_p$,…
take008
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8
votes
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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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…
Jos van Nieuwman
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7
votes
1 answer
Algorithm to find relations between polynomials
Let $p_1,\ldots,p_k\in\mathbb{C}[x_1,\ldots,x_n]$ be polynomials. Is there an algorithm to compute the ideal of relations between them?
More precisely, to find a set of generators for the kernel of the ring…
Simon Parker
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7
votes
2 answers
Irreducibility criteria for polynomials with several variables.
Let $K$ be a field. Show that $x^2-yz$ is irreducible in $K[x,y,z]$. Deduce that $x^2-yz$ is prime.
If it is $K[x]$, then there are several methods which can be used to check whether a given polynomial is irreducible. But how do we check that when…
Extremal
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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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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…
ThCastro
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6
votes
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 $x^3-z$.…
beertje00
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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…
Travis62
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6
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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