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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".

28
votes
2answers
16k views

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 $m \in \mathbb{N}$ ...
8
votes
2answers
65 views

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 ...
7
votes
3answers
438 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 ...
6
votes
2answers
66 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
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 $\mathbb{Q}[x, y]$. ...
6
votes
1answer
117 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 ...
5
votes
3answers
198 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 ...
5
votes
2answers
67 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], \...
5
votes
1answer
145 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,...
4
votes
2answers
392 views

Counting irreducible polynomials over finite fields [duplicate]

How many irreducible polynomials in $Z_2[x]$ of degree $3$? I have discussed this with my friend before and we found that $x^3 + x^2 + 1$ and $x^3+x+1$ are the two said polynomials which irreducible....
4
votes
2answers
75 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 $\varphi:\mathbb{Z}_5[x]\to\mathbb{Z}_5/(x^2+x+2)$ ...
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 ...
4
votes
2answers
365 views

Ideal of a Polynomial ring $R$ which is not principal.

Let $R$ be the ring given by $R=\mathbb Z+x\mathbb Q[x]$. Then show that: 1) $R$ is an integral domain and its units are $+1$ and $-1$. 2) $x$ is not prime in $R$ and describe the quotient ...
4
votes
2answers
53 views

What are all the isomorphisms from $\mathbb{K}[x] \to \mathbb{K}[x]$?

Let $\mathbb{K}$ be a field. Show that the $\mathbb{K}-$isomorphism $\mathbb{K}[t]\to \mathbb{K}[t]$ is given by $t\mapsto t-a.$ I have shown that it is an isomorphism as, Let $\psi : \mathbb{K}[t]\...
4
votes
2answers
1k views

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 ...
4
votes
2answers
70 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 ...
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 ...
4
votes
0answers
28 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,\...
4
votes
0answers
245 views

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 $R$. ...
3
votes
2answers
159 views

A Field of Polynomials of a Linear Operator

Consider a finite-dimensional vector space $V$ over the field $\mathbb{F}.$ Consider a linear operator $T : V \to V$ such that no nonzero subspace of $V$ is mapped into itself by $T.$ Let $\mathbb{F}[...
3
votes
1answer
2k views

Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain. [duplicate]

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 ...
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\...
3
votes
1answer
304 views

Prove that the field $k(x)$ of rational functions over $k$ in the variable $x$ is not a finitely generated $k$-algebra.

I am working through Chapter 15 of Dummit and Foote's Abstract Algebra text, and I am stumped on how to prove the following (Exercise 3): Prove that the field $k(x)$ of rational functions over $k$ in ...
3
votes
1answer
84 views

Describing a $\mathbb{Z}$-algebra and its tensor with $\mathbb{Z}/2\mathbb{Z}$

I am working with the $\mathbb{Z}$-algebra generated by the two elements $\alpha$ and $\beta=(\alpha^3+1)/2$ of $\mathbb{Q}(\alpha)$ such that $$\alpha^4+5\alpha^3+12\alpha^2+21\alpha+21=0.$$ I guess ...
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.
3
votes
1answer
29 views

notation in congruence relation

hi there i was looking through my lecture notes and i'm struggling to understand a particular piece of notation the vertical line | and i was wondering if you could explain its meaning $$f \sim g \...
3
votes
1answer
92 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 ...
3
votes
1answer
47 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]/...
3
votes
0answers
74 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-...
3
votes
0answers
47 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 ...
3
votes
0answers
34 views

Prime ideals in polynomial rings satisfying a particular condition

In the paper "Kronecker function rings and flat $D[X]$-modules" by Arnold and Brewer it's proved the following result: Lemma: Let $D$ be an integral domain. If $Q$ is a prime ideal of $D[X]$ such ...
3
votes
0answers
512 views

Polynomial rings over a commutative ring with identity

Let $R$ be a commutative ring with identity. Consider the polynomial ring in $n$ indeterminates $R[X_{1},\ldots ,X_{n}]$. Let $R[Y_{1},\ldots ,Y_{m}]$ be a polynomial sub-ring of it in $m$ ...
2
votes
3answers
80 views

Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that the $\gcd$ of $x^5$ and $x^6$ does not exist

Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that a gcd of $x^5$ and $x^6$ does not exist in $R$. I am reasoning for the absurd and I am ...
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, ...
2
votes
2answers
132 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 ...
2
votes
2answers
234 views

Defining irreducible polynomials over polynomial rings

Let R be a ring, and R[x] be a polynomial ring. Can we define what it means for a polynomial $p(x) \in R[x]$ to be irreducible over R[x]? Various sources (such as Wikipedia) only provide such ...
2
votes
2answers
84 views

Is the gcd of two polynomials in $F[x]$ where $F$ is a field unique?

I'm guessing not since it's not true for the integers (we have a negative and positive 'gcd', and we choose the positive one) I'd appreciate some examples because I'm really new to fields and what-...
2
votes
1answer
46 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 $...
2
votes
2answers
91 views

How to write polynomial rings.

I am recently struggling with a question presented to me by a friend: I need to write: $2 + xy − x^2y + 3x^2y^2 + 8x^2y^4 − x^3y ∈ Z[x, y]$ as an element in $Z[x][y]$ and as an element in $Z[y][x]...
2
votes
1answer
62 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 ...
2
votes
1answer
51 views

Noether polynomial ring over $2\Bbb Z$

Given a definition of $R=2\Bbb Z$ prove that ring $R[x]$ is not noether. I assume that the proof should be based on the fact that a ring is noether if and only if any ideal is finitely generated. ...
2
votes
2answers
70 views

Ideals in a polynomial ring

(Sorry for the title being so vague, as I didn't know how to summarize it) I have to solve the following problem: Let $F=P_{1}^{e_{1}}\cdots P_{r}^{e_{r}}$ be the prime decomposition of a ...
2
votes
2answers
68 views

Trying to prove that $\langle y-x^2, z-x^3\rangle$ is prime in $K[x,y,z]$

Let $K$ be a field. I am trying to prove that $I=\langle y-x^2, z-x^3\rangle$ is a prime ideal in $K[x,y,z]$. Main idea: Define a homomorphism $\phi:K[z,y,z]\rightarrow K[x]$ as $$\phi(x)=x, \phi(y)...
2
votes
1answer
77 views

Example for $c(fg)\neq c(f)c(g)$ [duplicate]

Let $R$ be a ring and let $f(X)$ be a nonzero polynomial in $R[X]$. The content of $f$ is the ideal $c(f)$ generated by the coefficients of f. The ring is called Gaussian if $c(fg)=c(f)c(g)$ for all ...
2
votes
2answers
148 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 ...
2
votes
1answer
48 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
2answers
376 views

Determining whether $(x^2-3)$ is a maximal ideal in $\mathbb{Z}[X]$

I've got a question regarding abstract algebra and prime/maximal ideals. I need to determine whether $(x^2-3)$ is a maximal or prime ideal in $\mathbb{Z}[X]$. I have not yet been introduced to ...
2
votes
1answer
65 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
votes
1answer
85 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
80 views

Identifying the ring $R=\mathbb{Z}_{9}[x]/(x^2-3,3x)$

I want to know if there is a simple form of the ring $$R=\mathbb{Z}_{9}[x]/(x^2-3,3x)$$ I tried to start with the equations $3x\equiv0$ and $x^2\equiv 3$. So, $3x^2\equiv 0$ and $3x^2\equiv 9$. ...