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]
411 questions
51
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
- 5,003
22
votes
2 answers
Division algorithm for multivariate polynomials?
We know that if $F$ is a field and $f(X)$ a non-zero polynomial in $F[X]$, then for every polynomial $g(X)$ we can find $q(X),r(X)$ such that
$$g(X)=f(X)\cdot q(X)+r(X)$$
with $r(X)$ the zero polynomial or $\deg r(X)<\deg f(X)$.
My question is: the…
Federica Maggioni
- 8,632
14
votes
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
- 121
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11
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
- 186,215
11
votes
1 answer
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
- 181,922
11
votes
3 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
- 5,945
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
- 233
9
votes
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
- 732
8
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
- 4,421
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
- 9,245
8
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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8
votes
0 answers
Gauss' lemma for arbitrary integral domains
One of the versions of the classical Gauss' lemma in abstract algebra states the following:
Theorem: Let $R$ be an integral domain, $f\in R[X]$ of positive degree and $K$ the quotient field of $R$. If $R$ is a UFD, these facts are equivalent:
i)…
Xam
- 6,338
7
votes
0 answers
Does the McCoy property pass to subrings?
A ring $R$ is said to be right McCoy if whenever $fg=0$ for nonzero $f,g\in R[x]$, there exists $r\ne 0$ in $R$ such that $fr=0$. Left McCoy rings defined similarly. A ring that is both left and right McCoy is called McCoy.
Some ring properties…
Maths Wizard
- 839
7
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
- 194
- 7
7
votes
4 answers
Gauss's Lemma Proof
Theorem 2.39 (Gauss’s Lemma). A polynomial $f ∈ \mathbb{Z}[x] ⊆ \mathbb{Q}[x]$ of the form $$f(x) = x^n + a_{n−1}x+^{n−1}+ ...+ a_1x + a_0$$ is irreducible in $\mathbb{Q}[x]$ if and only if it is irreducible in $\mathbb{Z}[x]$. More precisely, if…
MathGeek1998
- 279