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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".

3
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1answer
46 views

Ideals in polynomial ring

I am facing this problem: Prove that there is a bijection between the monic divisors of $x^n−1$ in $F[x]$ and the ideals of $F[x]/\left<x^n−1\right>$. I tried to find how the ideals in $F[x]/...
1
vote
0answers
12 views

Clarification about a proof regarding a sum of polynomials being expressed as a linear combination of S-polynomials

I'm reading this proof from Ideals, Varieties, and Algorithms by David A. Cox, Donal O'Shea, and John Little. You can find an online version here. This is Lemma 5 of Chapter 2, page 85. From my ...
0
votes
0answers
14 views

How can I simplify $\mathbb{F}[X,Y]/(X^2-Y^2)$?

Here $\mathbb{F}$ is an arbitrary field. $$\psi :\mathbb{F}[X,Y]\rightarrow\mathbb{F}[X]\times\mathbb{F}[X];\\ f\mapsto (f \ \text{mod} (X+Y),f\ \text{mod} (X-Y))$$Is a ring homomorphism with kernel $...
0
votes
2answers
30 views

Listing all Possible ideals of ring $F[x]/(p(x))$ where F is field and p(x) is polynomial in $F[x]$

I wanted to list all Possible ideals of ring $F[x]/(p(x))$ where F is field and p(x) is polynomial in $F[x]$ I can list ideal but I do not know my list contain all possible . My List of ideal $F[x]...
3
votes
1answer
86 views

Is $A[[x_1,x_2,\dots]]$ flat over $A[x_1,x_2,\dots]$? ($A$ noetherian and commutative)

Let $A$ be a noetherian commutative ring with one and $x_1,x_2,\dots$ indeterminates. Question. Is $A[[x_1,x_2,\dots]]$ flat over $A[x_1,x_2,\dots]\ ?$ Recall that $A[[x_1,x_2,\dots]]$ is the set ...
1
vote
3answers
29 views

Difference between $F[x,y] $ and $F(x)[y]$ in Ring theory

I come across following ring $F[x,y] $ and $F(x)[y]$ where $F$ is a field. I think both are same initially . But as $xy$ is irreducible in $F(x)[y]$ but reducible in $F[x,y] $. Which is a ...
2
votes
0answers
46 views

Prime ideals in multi-variable polynomial rings over $\mathbb{Z}$

There is a nice classification of prime ideals in the ring $\mathbb{Z}[x]$, see this question. Is there any generalization of this result, on $\mathbb{Z}[x_1,\cdots,x_n]$? Due to this post, I ...
1
vote
1answer
42 views

Kernel of a polynomial ring homomorphism

Let $\mathbb{F}$ be a field, and define a homomorphism $\phi:\mathbb{F}[x,y]\rightarrow \mathbb{F}[z]$ by: $$f(x,y)\mapsto f(z^a,z^b)$$ where $a\neq b\in \mathbb{N}$. My question is: For what $\...
0
votes
0answers
24 views

Surjections of polynomial rings, Krull dimension, and regular sequences

Let $k$ be a commutative Noetherian ring with unity (I will ultimately want $k = \mathbb Z$) and $$\varphi\colon A = k[x_1,\ldots,x_n] \to k[y_1,\ldots,y_p] = B$$ a surjective $k$-algebra homomorphism....
0
votes
0answers
6 views

Hilbert polynomials in two variables with Macaulay2

In J. Symb. Comput. (1999) 28, 681-710, Levin worked with bifiltered, finitely generated $R$-modules ($R$ being a polynomial ring in two sets of variables) and he found an analogue of the Hilbert ...
-1
votes
1answer
28 views

Describe explicitly $\text{spec}(\mathbb{R}[x])$ and $\text{spec}(\mathbb{C}[x])$ [closed]

Describe explicitly $\text{spec}(\mathbb{R}[x])$ and $\text{spec}(\mathbb{C}[x])$, (where for a given ring $R$, $\text{spec}(R)$ is defined to be the set of all prime ideals of $R$). I don't have an ...
4
votes
2answers
66 views

Zero divisors of $\mathbb{Z}_{7}[x] / (x^4+x^3-3)$ and inverse element of $\overline{x+1}$

How many zero divisors there are in the ring $\mathbb{Z}_{7}[X] / (x^4+x^3-3)$? What is the inverse element of $\overline{x+1}$? I'm not sure where to begin, so I thought it might be a good idea ...
5
votes
2answers
64 views

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], \...
2
votes
2answers
59 views

Product of ideals in $\mathbb{Z}[X]$

Consider the ideals $I = (2,X), J = (3,X) \in \mathbb{Z}[X]$. I want to show that the 'product set' $\Pi := \{ij \mid (i,j) \in I \times J\}$ is not an ideal in $\mathbb{Z}[X]$ and in particular, ...
0
votes
1answer
35 views

Subring of polynomial ring containing two polynomials of relatively prime degree

Let $R$ be an integral domain , $S$ be a subring of $R[x_1,..., x_n]$ . If $S$ contains two polynomials of relatively prime degree, then how to show that there exists an integer $m>1$ such that $...
4
votes
1answer
65 views

Does there always exist a linear shift of a given polynomial such that all coefficients are nonzero?

Let $K$ be a field and let $f(x)=a_0+a_1x+\cdots +a_nx^n\in K[x]$ be such that $a_n\neq 0$. Does there always exists some $\alpha\in K$ such that the coefficients of $f(x-\alpha)$ are nonzero in every ...
2
votes
1answer
32 views

Showing if $R$ is a PID and $0\neq r\in R$ is irreducible, then $R[X]/(r)\cong (R/(r))[X]$

I've been looking at the statements I found on these two Stack Exchange answers and I've been trying to prove them. The first claim is: If $R$ is a PID and $0\neq r\in R$ is irreducible, then $R[X]/...
1
vote
0answers
45 views

$\mathbb{ R } \left[ x _ { 1 } , \ldots , x _ { n } \right] / \left( x _ {1} ^ {2} + \cdots + x _ {n} ^ {2} - 1 \right)$ is a UFD when $n>2$

Prove $$\mathbb { R } \left[ x _ { 1 } , \ldots , x _ { n } \right] / \left( x _ { 1 } ^ { 2 } + \cdots + x _ { n } ^ { 2 } - 1 \right)$$ (i) is not a UFD when $n=2$ (ii) is a UFD when $n&...
0
votes
1answer
28 views

Applying degree 2 or 3 irreducibility tests to higher degree

Given the polynomial$\ x^4+x+1$, I have to find out if it is irreducible over $\mathbb Q $. When looking at the solutions, they applied the degree 2 or 3 irreducibly tests to determine that it ...
3
votes
2answers
58 views

Find all of homomorphisms $Φ$ from $\mathbb C$[$x$]/$I$ to $\mathbb C$ that satisfies specific condition

I want to know how can I find ALL homomorphism that satisfies the condition mentioned in the question above. Please give me a method or answer in detail.
1
vote
2answers
34 views

GCDs for the polynomial ring over a Galois field.

You can find many examples of computing the inverse of an element inside a Galois field. (For example here) What happens if we look at the polynomial ring over a Galois field and would like to ...
0
votes
2answers
58 views

coprime ideals in $K[X]$

If $K$ is a field, $A=K[X]$, take $m,n \in K$ such that $m \ne n$. Prove that the ideals $I=(X-m)$ and $J=(X-n)$ are coprime. I know the regular definition of coprime. But here, should we prove $I + ...
1
vote
1answer
25 views

Defining an evaluation map between $\mathbb{Q}\lbrack x \rbrack$ and $\mathbb{Q}\lbrack x \rbrack/(x^2-5)$

I need to define an evaluation mapping between $\mathbb{Q}\lbrack x \rbrack$ and $\mathbb{Q}\lbrack x \rbrack/(x^2-5)$. I know I want the identity to map to the identity, but I'm not sure what the ...
0
votes
0answers
24 views

Heuristics for finding minimal primes/ primary decomp. Example: (XZ-Y^2, X^3-YZ)

In occassions where I need to find the minimal primes associated to an ideal / find a primary decomp., sometimes I can do it just fine, sometimes I find myself completely blind in the search for it. ...
0
votes
0answers
40 views

Norm on k[X] or Q[x]?

While considering specific examples of norms on number fields, I was considering $\mathbb{Q}[\sqrt{a}]\cong \frac{\mathbb{Q}[x]}{(x^2-a)}$. This led me to the following question: Given a field $k$, ...
0
votes
2answers
80 views

Is there an efficient algorithm to find all zeros of systems of multivariate polynomial equations over a finite field?

I want my computer to solve large systems of multivariate polynomial equations over a finite field. The field is $\mathbb F_p$, where $p$ is a prime number. I heard that there is an algorithm using ...
1
vote
1answer
50 views

Please help me understand “The Weak Nullstellensatz.”

I'm studying Lovett. "Abstract Algebra." Chapter 12. Multivariable Polynomial Rings. Section 3. The Nullstellensatz. The book deduces The weak nullstellensatz from proposition 12.3.2. The author ...
0
votes
1answer
63 views

What does it mean by that $E$ is a finite extension of $F$, when $F\subseteq E$ is not clear?

I'm studying Lovett. "Abstract Algebra." Chapter 12. Multivariable Polynomial Rings. Section 3. The Nullstellensatz. I don't understand the exact meaning of the following proposition. Proposition ...
6
votes
1answer
44 views

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 $\mathbb{Q}[x, y]$. ...
1
vote
2answers
48 views

Counterexample PID [duplicate]

We know that if F is a field, then the polynomial ring over F is a PID. Do you have a counterexample that shows that if F isn’t a field than the polynomial ring over F isn’t a PID?
3
votes
1answer
50 views

A computational criterion of irreducibility in $\mathbb Z[X]$?

If $f\in\mathbb Z[X]$ and there are $x_1,\dots, x_n\in\mathbb Z_+$, where $n>\deg f$, such that $f(x_i)\in\mathbb P$, $i=1,\dots n$, then $f$ is irreducible over $\mathbb Z$. Because, if $\,f=g\...
0
votes
0answers
30 views

Field of fractions and rational functions

When I have a field $k$ and take the ring of polynomials in the variables $x1, x2, ..., xn$, and subsequently take the quotient field of these polynomials, I was asking myself, is this in 1:1 ...
0
votes
2answers
159 views

Finding units and zero divisors in a polynomial quotient ring

I am trying to study for an exam and I am not sure of this solution my professor posted to an exercise. I am given the polynomial quotient ring $\mathbb{Z}_6/(x^2+2x)$ and have to find all units and ...
0
votes
2answers
45 views

Prove $(0) = (x)\cap (xz^{n-1} + \lambda y^n)$ in $R=\frac{k[x,y,z]}{(x^2,xy)}$

Studying for my algebra final and doing some practice problems, and I can't seem to understand this one... Full problem: Let $k$ be a field, and $R=\frac{k[x,y,z]}{(x^2,xy)}$. For $n\in\mathbb{N}, \...
1
vote
2answers
49 views

How the following multiplication table is solved ( related to $F_2[X]/f(x)$ )

$F_2$ is polynomial field of group of integer modulo $2.f(x)$ is $x^2 + x + 1$. I didn't got how the multiplication is happening in the table.I referred to many sources related to this topic but ...
2
votes
0answers
33 views

Cyclic multiplicative subgroups

Let $R[t]$ be the polynomial ring over the nonzero commutative ring $R$ and $f_R$ be the associated polynomial function. If $|\{\alpha \in R : f_R(\alpha)=0\}| \leq \deg(f)$ for every $0 \neq f \...
1
vote
0answers
39 views

Isomorphism between quotient fields of polynomial rings [closed]

Let $R$ be an integral domain and $F$ be its field of fractions. If $X$ is a nonempty set, prove that there is a ring monomorphism from $R[X]$ to $F[X]$ that extends to an isomorphism of their ...
1
vote
1answer
46 views

Homomorphisms and automorphisms on polynomial rings

I am trying to prove a series of propositions: Given any homomorphism p from $\mathbb{R}$[X] to $\mathbb{R}$[X], show that it is equal to $\phi_g$ for a unique g in $\mathbb{R}$[X], with $\phi_g$(f) =...
2
votes
0answers
42 views

Classifying homomorphisms on polynomial rings with real coefficients.

Show that every homomorphism $\mathbb{R}$[X] $\rightarrow$ $\mathbb{R}$[X] can is equal to $φ_g$ for a unique g $\in$ $\mathbb{R}$[X], given by $φ_g(f)$ = $f(g(X))$ My guess for any homomorphism $h$, ...
1
vote
0answers
17 views

Answer verification to question about polynomial rings and derivatives.

Can someone check whether my answers are okay to the following Judson question? (a) Let $f(x)=a_{n}x^{n}+\cdots +a_{m}x^{m}+\cdots a_{1}x+a_{0}$ and $g(x)=b_{m}x^{m}+\cdots +b_{1}x+b_{0}$. Assume ...
2
votes
1answer
75 views

Fundamental ismorphism theorem

I don't understand how to apply the fundamental isomorphism theorem to polynomial quotient rings. For example is the ring $\mathbb C[X,Y,Z]/\langle X^2-Z,XZ-Y^3\rangle$ isomorphic to $\mathbb C[X,Y]/\...
-1
votes
2answers
54 views

Prove or Disprove $G(x, y)=G(y,x)$ [closed]

Let $G(x, y)$ be a polynomial such that: $$\frac{\partial}{\partial x}G(x, y)=\frac{\partial}{\partial y}G(x, y)$$ Prove or disprove that $G(x, y)=G(y,x)$.
5
votes
1answer
139 views

Is $(x^2+y^2-1, z^2+w^2-1)$ a prime ideal in $\mathbb Q[x,y,z,w]$?

$\newcommand{\Q}{\mathbb Q}$ I saw an argument that the ideal $I=(x^2+y^2-1, z^2+w^2-1)$ is a prime ideal in $\Q[x,y,z,w]$ but I cannot see why. I tried to find a surjective homomorphism from $\Q[x,y,...
0
votes
1answer
37 views

If $f|d$, then $f$ is invertible in $R[X]/(d)R[X]$

Let $R$ be a field and $f$ and $d$ be polynomials in $R[X]$. If $f|d$, then $f$ is invertible in $R[X]/(d)R[X]$. I tried to either prove of disprove this statement, but so far I haven't been able ...
1
vote
1answer
24 views

No element of degree $0$ (constant $= a_0 X^0$) in $\mathbb{Z}_4/(2X) \mathbb{Z}_4$

Consider $\mathbb{Z}_4[X]/(2X) \mathbb{Z}_4[X]$. Then we wish to show/verify that the residue class $X$ does not contain an element of degree $0$. In the previous exercise the book asked that if we ...
0
votes
2answers
30 views

$a^4=0$ then $1-a$ is invertible in $R[x]/(d)R[x]$ [duplicate]

Suppose $a^4=0$, for some $a \in R[x]/(d)R[x]$, then prove that $1-a$ is invertible. I was thinking since $a^4 = a \cdot a \cdot a \cdot a=0$, this implies that $a$ has to be zero (?) . Now we have ...
1
vote
1answer
51 views

The maps $f_+$ and $f_-$ are well-defined.

We define the maps $f_+$ and $f_-$ from $\mathbb{Q}[X]/(X^2 -2) \mathbb{Q}[X]$ to $\mathbb{Q} + \mathbb{Q} \cdot \sqrt2$ in the following manner: For any residue class $g+ (X^2-2) \mathbb{Q}[X]$, we ...
1
vote
1answer
43 views

Prove $x^4 +x+1$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[x]$ [duplicate]

$ x^4 +x+1$ in $\mathbb{Z}/2\mathbb{Z}[x]$ is an irreducible polynomial. So far we have only treated quadratic and cubic polynomials, which are irreducible if they do not have any zeros. However, ...
3
votes
0answers
72 views

determining if quotient ring of polynomials over a finite field is a field or not

I am stuck with this question: "Determine if $GF(2011^2) [x] /<x^4-6x+12>$ is a field or not." I know that since this polynomial ring is defined over a field, I only have to determine if $x^4-...
1
vote
1answer
41 views

What are the polynomial quotients $R/x$ and $R/(R/x)$ for $R = (\mathbb{R}[x]/x^n)$?

Define the polynomial ring quotient $R = \mathbb{R}[x]/x^n$. Is my understanding correct that $$ R/x \cong \mathbb{R} $$ and accordingly, as scalars divide all polynomials, $$R/(R/x) \cong \{1\}$$ ...