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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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Towering of ideals

Is it true that $\mathbb{Z}[x,y]/(p,q) \simeq (\mathbb{Z}[x,y]/(p))/(q)$, where $p \in \mathbb{Z}[x]$ and $q \in \mathbb{Z}[y]$?
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1answer
23 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
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2answers
54 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
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2answers
30 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
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2answers
47 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 + ...
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1answer
22 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
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0answers
21 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. ...
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0answers
34 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 $k$, ...
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2answers
72 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
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1answer
38 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 ...
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1answer
58 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 ...
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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 $\mathbb{Q}[x, y]$. ...
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2answers
44 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?
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1answer
46 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\...
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0answers
27 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
66 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
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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
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2answers
41 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 ...
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0answers
27 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
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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 ...
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1answer
42 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
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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
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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
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1answer
72 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]/\...
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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 $G(x, y)=G(y,x)$.
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1answer
131 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,...
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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
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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
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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 ...
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1answer
49 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 ...
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1answer
42 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
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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-...
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1answer
35 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
34 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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2answers
47 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
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1answer
44 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 ...
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0answers
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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 ...
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2answers
74 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 $\varphi:\mathbb{Z}_5[x]\to\mathbb{Z}_5/(x^2+x+2)$ ...
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0answers
40 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
114 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 ...
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2answers
64 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 ...
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1answer
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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 ...
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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 ...
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1answer
63 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
75 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
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1answer
44 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
45 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
46 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}...
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3answers
317 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 ...