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".
411 questions
0
votes
1
answer
97
views
What would be maximal ideal of $K+xL[x]$?
Let $K$ be a subfield of $L$, where $K$ is algebraically closed. We know that maximal ideals of $K[x]$ are exactly of the form $\langle x-a \rangle$, such that $a\in K$.
Can we deduce that maximal ...
- 81
0
votes
0
answers
49
views
Linear factors of polynomial in a polynomial ring
What is the fastest known method to split $f \in \mathbb{Z}[x]$ into linear factors mod $g \in \mathbb{Z}[y]$, assuming this happens?
Since that is not very clear, here is an example of what I'm ...
- 335
1
vote
1
answer
62
views
Associated polynomial function vs evaluation map
In Bosch's "Algebra: From the Viewpoint of Galois Theory" (page 29), the author considers a ring extension $R\subset R'$, a polynomial $f=\sum_ia_iX^i\in R[X]$, and says that we can ...
- 1,308
0
votes
0
answers
60
views
For each $ n \in \mathbb{N} $, find an integral domain with a finite number of units that has Krull dimension $ n $.
For each $ n \in \mathbb{N} $, find an integral domain $A$ with a finite number of units and $\dim_{\text{krull}}A=n$.
I was thinking about $A=K[x_1,...,x_n]$ with $n \in \mathbb{N}$ and $K$ a field ...
- 193
1
vote
0
answers
134
views
Unique automorphism of $R[x]$
I'm basically stuck on a particular question from the Hungerford book about rings, and specifically, an automorphism of rings.
The question is:
Let $R$ be a commutative ring with identity, and let $c, ...
- 53
0
votes
0
answers
29
views
Modular System for polynomial Rings
i am wondering if there is a nice modular system for polynomial Rings over the integers or a field, to use the chinese remainder theorem.
For the integers we have
$$\mathbb{Z}_n \simeq \prod \mathbb{Z}...
- 31
7
votes
0
answers
120
views
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 ...
- 845
0
votes
1
answer
67
views
Let $F$ be a field, and $f(x), g(y)$ be nonconstant polynomials in $R = F[x,y]$. Show that the ideal $I = (f(x), g(y))$ is not all of $R$
I'm looking for help on the title problem.
The ideal $I = (f(x), g(y))$ is only equal to all of $R$ if $1 \in I$. So suppose
$1 = a(x,y)f(x) + b(x,y)g(y) \in I$
I've played around with this equation a ...
- 61
0
votes
0
answers
60
views
Edge ideal of a non-simple graph
A graph $G$ is a triple $(V,E,\varphi)$ where $V$ and $E$ are sets, $\varphi$ is a function with domain $E$ and with range $\mathscr{P}_{\leq2}(V)$ (the set of subsets of $V$ with at most $2$ elements)...
- 4,169
2
votes
1
answer
118
views
Polynomial ring quotient by an ideal generated by an irreducible polynomial
Question: Suppose that a commutative ring $E$ containing $R$ as a subfield with the same unity and an element $a \in E$ is a zero of an irreducible polynomial $r(x) \in R[x]$. Is $ R[x]/\langle r(x) \...
- 1,973
1
vote
0
answers
130
views
Possibly false statement of Exercise 5.16 of Atiyah and Macdonald
Question is originated from this post of mine. Following is statement of Noether normalization lemma given in Atiyah and Macdonald:
Let $k$ be an infinite field and let $A \neq 0$ be a finitely ...
- 4,645
2
votes
0
answers
62
views
Regularity of $R / (I+J)$ in terms of the regularity of $I$ and $J$
I am studying the Castelnuovo-Mumford regularity (simply said regularity) function and have the following situation:
Let $ I $ and $ J $ be two graded ideals in $ R = K[X] $, where $ X = \{x_1, \dots, ...
user969954
-1
votes
2
answers
138
views
Example of a polynomial of degree $n$ that has at more than $n$ roots [closed]
Context: Here is the problem of my professor give to us:
We all know that a polynomial of degree $n$ in $ \mathbb{C} $ has at most n roots, but is there any ring $A$ such that a polynomial of degree $...
- 29
-2
votes
1
answer
104
views
Does height of an ideal and collection of minimal primes over that ideal remain invariant under extension of polynomial ring?
Let $I$ be an ideal in $K[x_1,...,x_k]$ where $K$ is a field and $I_1 = (f_1,..,f_t)$ where $f_i$'s are polynomial in $K[x_1,...,x_k]$.
Now considering
the same $f_i$'s as elements of $K[x_1,..,x_n]$ ...
- 87
0
votes
0
answers
36
views
Characterising maximal ideals in polynomial rings over Noetherian rings.
Let $R[\vec{x}]=R[x_1,x_2,...,x_n]$ be a polynomial ring over a UFD with the Noetherian property. The following is an attempt to characterise its maximal ideals.
Since $R$ is Noetherian and a UFD, ...
