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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".

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2answers
32 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
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1answer
33 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
21 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 ...
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2answers
27 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 ...
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2answers
34 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
31 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
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0answers
25 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 \...
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0answers
19 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
41 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) =...
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0answers
21 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 $h$, ...
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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
69 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
51 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 $G(x, y)=G(y,x)$.
6
votes
1answer
114 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
36 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
23 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
28 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
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1answer
47 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
38 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, ...
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0answers
68 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
28 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, $$R/(R/x) \cong \{1\}$$ ...
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0answers
26 views

$\gcd(x,y)=1$ then $\gcd(c \cdot x, c \cdot y)=c$ for polynomials

As a part of a proof I want to use that for polynomials $c, x,y$ it is the case that $\gcd(x,y)=1$ then $\gcd(c \cdot x, c \cdot y)=c$. Is this always true?
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vote
2answers
39 views

For polynomials $f, g$ and $\gcd(f(X),g(X))=d(X)$ then $\gcd(f(X+a),g(X+a))=d(X+a)$

Suppose for polynomials f, g in $\mathbb{Q}[X]$ it holds that $$\gcd(f(X),g(X))=d(X)$$ What we also want to now prove is that for $a \in \mathbb{Q}$: $$\gcd(f(X+a),g(X+a))=d(X+a)$$ So the ...
2
votes
1answer
39 views

How many polynomials of degree $n$ in $\mathbb{Z}/3\mathbb{Z}$

We are given the polynomial ring $\mathbb{Z}/3\mathbb{Z}$. We are asked to determine how many polynomials of degree $n$ there are. First of all, the possible coefficients are $0, 1,2$, if we have a ...
2
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0answers
33 views

Prove that any polynomial with integer coefficients must have a composite number in its image. [duplicate]

Let $f \in \mathbb{Z}[x]$ be a polynomial of degree at least $1$. Prove that there is $n \in \mathbb{Z} $ such that the corresponding polynomial function $f(n)$ is not a prime. I think that the ...
4
votes
2answers
72 views

Show that $\mathbb{Z}_5[x]/(x^2+x+1)\cong\mathbb{Z}_5[x]/(x^2+x+2)$

I think this problem is from Gallian, prof couldn't solve it. Notice that both polynomials have no roots. I tried to construct an onto homomorphism $\varphi:\mathbb{Z}_5[x]\to\mathbb{Z}_5/(x^2+x+2)$ ...
3
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0answers
38 views

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 ...
2
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2answers
82 views

Given any commutative ring $R$ with unity, $R[X]$ has infinitely many maximal ideals.

Hope this isn't a duplicate. I was trying to answer the following questions: (i) Let $k$ be any field. Then prove that $k[X]$ has infinitely many maximal ideals. (ii) Using (i) prove that, given ...
6
votes
2answers
63 views

$\Bbb R[X,Y]/(F) \cong \Bbb R[Z]$ 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 ...
6
votes
1answer
91 views

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 ...
0
votes
1answer
28 views

Polynomial Evaluation. How to (formally) substitute X with a Matrix

for simplicity a specific case: $f:=(a_i)_{i \in \mathbb{N}} \in \mathbb{R}[X]$ $A \in \mathbb{R}^{4\times 4}$ In one of our assignments f is evaluated as $\sum_{i=0}^{\deg(f)} a_iA^i$. How is ...
1
vote
1answer
62 views

Show that $\mathbb{Z}_3[i]$ is a field

I know that $\dfrac{\mathbb{Z}_3[x]}{\langle x^2+1\rangle}$ is isomorphic to $\mathbb{Z}_3[i]$, does this help me prove that $\mathbb{Z}_3[i]$ is a field? $\langle x^2+1\rangle$ is the ideal ...
2
votes
1answer
68 views

Show that every nonzero prime ideal in $\mathbb{C}[x, y]/(x^2 - y^2 - 1)$ is maximal [duplicate]

Show that the ring $A := \mathbb{C}[x, y]/(x^2 - y^2 - 1)$ is an integral domain. Further show that every nonzero prime ideal in A is maximal. I proved that $A$ is an integral domain by showing that $...
2
votes
1answer
43 views

How do the proper ideals of $\Bbb Q[x]/ \langle x^3 \rangle$ look like?

How do the proper ideals of $\Bbb Q[x]/ \langle x^3 \rangle$ look like? I know that the elements of it look like $p(x)$ modulo $x^3$. So the ideals of it look like $\langle ax^2+bx+c\rangle $ where $...
4
votes
1answer
44 views

Why isn't this criterion for determining irreducibilty working?

I have learned this criterion for irreducibility of polynomials: Let $R$ be an integral domain, let $I$ be a proper ideal of $R$, and let $p(x)$ be a non-constant monic polynomial in $R[x]$. If the ...
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0answers
45 views

Trouble with interpreting a question about polynomial rings?

I trying to do problem $9.3.1$ in Dummut & Foote; I think they use some "abuse of notation" that I don't understand. Let $R$ be an integral domain with quotient field $F$ and let $p(x)$ be a ...
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2answers
43 views

Trouble with a step in proof of "$R$ is a U.F.D. $\implies R[x]$ is a U.F.D.

I am reading a proof of this in Dummit & Foote (Chapter $9.3$, Thm. $7$). The proof uses Gauss' Lemma: Gauss' Lemma: Let $R$ be a U.F.D. and let $F$ be its field of fractions. If $p(x) \in {R}...
6
votes
3answers
263 views

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 ...
2
votes
2answers
67 views

If $R$ is a commutative ring with identity then $R[x_1, x_2, …, x_n]$ is not a P.I.D.?

I am working on problem $7$, section $9.1$ in Dummit & Foote: Let $R$ be a commutative ring with $1$. Prove that a polynomial ring in more than one variable over $R$ is not a principal ideal ...
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0answers
39 views

Relationship between $F(x, y)$ and $F[x, x^2y, x^3y^2, …, x^{n+1}y^n, …]$, is my understanding correct?

I am doing problem $8$ in section $9.1$ in Dummit and Foote: Let $F$ be a field and let $R = F[x, x^2y, x^3y^2, ..., x^{n+1}y^n, ...]$ be a subring of the polynomial ring $F[x, y]$. $a)$ ...
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votes
3answers
176 views

How do we know that automorphisms on polynomials have a polynomial like form?

Let $F$ be a field and $\sigma:F[x]\to F[x]$ be automorphism, $\sigma(a) = a$ for all $a\in F$. I'm supposed to show that $\sigma(f(x)) = f(ax+b)$ for some $a\not = 0$ and $b$ in $F$. Now I've got a ...
4
votes
0answers
27 views

Function $(f^n)_i:=\begin{cases}1\text{ if }i=n,\\0\text{ else }\end{cases}$ on polynomial ring $R[x]$

Let $R$ be a ring with $1$ and $R[x]$ be the polynomial ring in $x$ over $R$ with pointwise defined addition and convolution as multiplication. Let $f\in R[x]$ be $$f_i:=\begin{cases}1\text{ if }i=1,\...
0
votes
2answers
37 views

Finite field adjunction - isomorphism that maps from the field into a polynomial ring factored over an ideal

A field $(K, +, 0, *, 1)$ is given, and it is a finite field with $25$ elements. $P$ is the prime field over K. Further it was given that a polynomial $r(x):=x^2+x+1 \in P[x]$ is irreducible. Now, ...
0
votes
1answer
31 views

Surjective ring homomorphism - cardinality of the domain

I've been doing some exam preparation examples, and I wanted to ask if my idea to this example makes sense - at least if I am on the right path. A field $P \cong \mathbb{Z}_7$ and a polynomial ring $...
1
vote
2answers
54 views

Determine all the ideals of $\mathbb{Q}[x,y]$ that contain the ideal $\langle x^2,y^2,xy\rangle$.

Determine all the ideals of $\mathbb{Q}[x,y]$ that contain the ideal $I=\langle x^2,y^2,xy\rangle$. Are these all: $$\langle x,y\rangle$$ $$I$$ $$\mathbb{Q}[x,y]$$ ? EDIT.... $$\langle x^2,y \...
2
votes
1answer
45 views

Proving that $\mathbb{Z}[x]/(x^3-2x+1)$ is not an integral domain and that x is a unit

Proving that $E=\mathbb{Z}[x]/(x^3-2x+1)$ is not an integral domain and that $x$ is a unit. Let $r=(x^3-2x+1)$ So to show that $\mathbb{Z}[x]/(x^3-2x+1)$ is not an integral domain I must find a $p(x)...
2
votes
1answer
40 views

Let f (x) be a nonconstant element of Z[x]. Prove that f (x) takes on infinitely many values in Z.

This is a homework problem so no need to give me an immediate answer. My general plan has been to try to prove that f(x) cannot have both an upper bound and a lower bound. One strategy I tried was ...
1
vote
1answer
24 views

Ideal generated by $x_1 - a_1, \dots, x_k - a_k$ is prime in polynomial ring over integral domain

I would like to show the ideal $P_k = \langle x_1 - a_1, \dots, x_k - a_k \rangle$ of $R = F[x_1,\dots,x_n]$, where $F$ is a field, $k \leq n$ and $a_i \in F$, is a prime ideal. In the case where all ...
0
votes
0answers
29 views

Help with fixing proof that transcendental extension of $\mathbb{Q}$ is dense in transcendental extension of $\mathbb{Q}_{p}$

Good morning. I am trying to prove the following statement (I suspect it is true): Consider a finite set $(a_{1},...,a_{n})$ of elements in an elementary extension $*\mathbb{Q}_{p}$ of $\mathbb{Q}_{p}...
0
votes
0answers
26 views

numerical calculation of principal ideal generator in polynomial ring

I will first state my problem and explain where I am at with the problem. Let $I$ be a principle ideal generated by a single, multivariate polynomial $p \in \mathbb{C}[z_1,z_2,z_3]$. I estimate a ...