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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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Ideal of multivariate polynomial

Let $p(x_1,...,x_n)$ be an element of $\mathbb{F}_2[x_1,...,x_n]/(x_1^2+1,...,x_n^2+1)$. Is there a way to capture the size or dimension of the ideal $(p(x))$? I would guess that if there is a way, it ...
user 1987's user avatar
  • 764
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Matrix of a module over a complex polynomial ring

I'm having trouble with the following question: Given a $\mathbb{C}[x]$ module of the form $M=\bigoplus_1^n\mathbb{C}[x]/(x-\lambda_i)^{n_i}$. what condition on the $\lambda_i$ ensures that $M$ is ...
Ben Carpenter's user avatar
0 votes
0 answers
22 views

Intersection of two ideals, <p(x)> and <q(x)> where p(x),q(x) are in a PID R[x] [duplicate]

I was guessing that <p(x)> intersection <q(x)> to be the ideal<LCM(p(x),q(x)>. But I couldn't proceed. Please help me to complete the proof if my intuition is correct. Otherwise give ...
Souvik Dolai's user avatar
0 votes
2 answers
27 views

Proving the equivalence of two fields which are quotients of polynomial rings

Apologies if this question has been asked in some form before, but I wasn't able to find it. Isomorphic quotient rings of polynomial rings over field is a similar question but it asks for the ...
lechatonnoir's user avatar
0 votes
2 answers
59 views

Given a commutative ring $R$ and an element $a \in R$ construct a new ring in which a becomes invertible

Hello all. I'm working my way through this problem for my algebra class and I'm currently having difficulty with part E. I've managed to solve every other part of this problem, most importantly D. ...
Norftopia's user avatar
3 votes
2 answers
84 views

Are these two groups $\mathbb{Q}[x]$ and $\mathbb{Q}$ under addition isomorphic or not?

I am a beginner at ring theory, and after studying the polynomial rings, I have known about the polynomial ring $\mathbb{Z}[x], \mathbb{Q}[x], \mathbb{R}[x]$ which forms a group under addition. After ...
RITAM SADHUKHAN's user avatar
6 votes
3 answers
143 views

prove that $x^{51}$ is congruent to $1\bmod(x^8+x^4+x^3+x+1)$

Consider $m(x)=x^8+x^4+x^3+x+1\in\mathbb{Z}_2[x]$ as used in AES. Next define the field $GL(2^8)=\mathbb{Z}_2[x]/m(x)$. How might I go about showing that $x^{51} \equiv1\bmod(m(x))$. I've tried ...
Tristan Hart's user avatar
0 votes
1 answer
27 views

Are solutions to polynomial equations subideals in a polynomial ring?

I'm wondering how can i make sense of a partial solution to a two-system of polynomial equations. Let $\{f,g\}\subseteq K[x,y]$ be polynomials in a polynomial ring over the field $K$. Given the ...
Simón Flavio Ibañez's user avatar
3 votes
0 answers
34 views

If $R$ for a commutative ring with identity (not necessarily an integral domain) satisfies ACCP, then does $R[x]$ also have ACCP?

I read here that if $R$ is an integral domain and has ACCP, then $R[x]$ also has ACCP. However, is this necessarily true for a commutative ring with identity? If it is false, then what is a ...
852619's user avatar
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1 vote
1 answer
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What does this sentence mean in polynomial rings?

I have trouble with understanding this sentence in the context of polynomial rings: "We will assume that polynomials satisfy right evaluation. That is, a polynomial can only be evaluated once it ...
Mahtab's user avatar
  • 733
5 votes
1 answer
80 views

Show that $\mathbb Q[t] \otimes_{\mathbb Q[t^2]} \mathbb Q[t] \cong \mathbb Q[x,y]/(x^2-y^2)$ (as $\mathbb Q$--algebras)

Show that $\mathbb Q[t] \otimes_{\mathbb Q[t^2]} \mathbb Q[t] \cong \mathbb Q[x,y]/(x^2-y^2)$ (as $\mathbb Q$--algebras) So I first tried to show this by first defining the map $\varphi: \mathbb Q[t] ...
Squirrel-Power's user avatar
2 votes
0 answers
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Let $R=\mathbb Z[X,Y]$. Find an exact sequence $0\to R\to R\oplus R \to R\xrightarrow{f} \mathbb Z \to 0$ with $f(g(x,y)) := g(0,0)$ for all $g \in R$

Let $R = \mathbb Z[X,Y]$. Construct an exact sequence of $R$-modules $$0 \to R\to R\oplus R \to R\xrightarrow{f} \mathbb Z \to 0,$$ where $f(g(X,Y)) = g(0,0)$ for all $g \in R$. Here $\mathbb Z$ is ...
Squirrel-Power's user avatar
1 vote
3 answers
103 views

How to show that $(y-x^2, z-x^3) \in \mathbb{C}[x,y,z]$ is irreducible or radical?

In Fulton's introduction to Algebraic Geometry, there is the following exercise on page 11: I have been struggling for a bit longer than I care to admit on this problem, and have not been able to get ...
Nikolas Koutroulakis's user avatar
1 vote
0 answers
43 views

Is there a way to understand the group of polynomials mod an ideal?

As a simple example: $$ \mathbb{Z}[n]/ \langle n^2 - an - b \rangle $$ We know this is simply all the linear expressions $\{cn+d \mid c,d \in \mathbb{Z}\}$ But is there an explicit way to describe the ...
user14448's user avatar
0 votes
0 answers
52 views

Factorization of $f(x)=x$ in $\Bbb{Z}/n\Bbb{Z}[x]$ where $n$ is the product of $k$ distinct primes (considering $\Bbb{Z}/30\Bbb{Z}[x]$ as example)

Intro This is the final part of the problem 9.4.20 (e), (d) from Dummit and Foote's Abstract Algebra. They formulate it as "Determine all the factorizations of $f(x)=x$ in $\Bbb{Z}/n\Bbb{Z}[x]$ ...
Stanarth's user avatar
  • 116
0 votes
1 answer
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Distributive property of sum and intersection of ideals in a polynomial ring [closed]

Let $S=k[x_1,\dots,x_n]$, where $k$ is a field, and $I,J,K$ be three ideals of $S$. Are the following right or are they wrong? $ I+ (J\cap K)=(I+J)\cap (I+ K)$ $ I\cap (J+ K)=(I\cap J)+ (I\cap K)$
Hola's user avatar
  • 151
2 votes
2 answers
61 views

If $I = (xy,(x-y)z)$ is an ideal in $k[x,y,z]$ where $k$ is a field, then $\operatorname{rad} I = (xy,xz,yz)$.

This is taken from Dummit & Foote, chapter 15.2, exercise 5. I am only quoting the part I am currently having trouble with: If $I = (xy,(x-y)z)$ is an ideal in $k[x,y,z]$ where $k$ is a field, ...
Ben123's user avatar
  • 1,104
2 votes
1 answer
66 views

Isomorphism between polynomial rings with several variables

Here is a problem form my abstract algebra course : Let $R = \frac{\mathbb{Q}[u,v,w]}{(u^2v^2 - w^3)}$. Find finitely many monomials $x^{a_j}y^{b_j}$ in $S = \mathbb{Q}[x,y]$ such that $R \simeq \...
EtiBeranger's user avatar
1 vote
1 answer
90 views

Explicit isomorphism between $R[X]/(x^2+1)$ and $R[x]/(x^2+3)$

The given question states that you have to proof that $\mathbb{R} [X]/(x^2+1)$ and $\mathbb{R} [x]/(x^2+3)$ are isomorphic, and then give an explicit isomorphism between them. I have already showed ...
MM3's user avatar
  • 53
2 votes
0 answers
100 views

Exhibit all the ideals in the ring $F[x]/(p(x))$, where $F$ is a field and $p(x)∈F[x]$ (describe them in terms of the factorization of $p(x)$)

This is Exercise 9.2.5 in Dummit and Foote's Abstract Algebra Exhibit all the ideals in the ring $F[x]/(p(x))$, where $F$ is a field and $p(x)$ is a polynomial in $F[x]$ (describe them in terms of ...
Stanarth's user avatar
  • 116
0 votes
1 answer
49 views

Finding the inverse of $x^2-x-2 \in \Bbb{Z}_5[x]/(x^3-2x^2+2)$

I'm a relatively new student to this topic, please give comprehensive explanations This is what I have so far: We can use Euclidean algorithm to solve the following and we have: $$\gcd(x^3-2x^2+2, x^2-...
Mengen Liu's user avatar
1 vote
0 answers
45 views

Determine if $(x^5+x^3y^2+x^2y^3+y^5-y)$ is a prime ideal in $\Bbb Q[x,y]$.

I am guessing $\mathfrak p=(x^5+x^3y^2+x^2y^3+y^5-y)$ is a prime ideal. Here is my attempt: Every $y^n$ term for $n\geq 5$ can be reduce to a lower degree modulo $\mathfrak p$ by using $$y^5 = y-x^2y^...
user108580's user avatar
1 vote
0 answers
88 views

Showing that $x^3 + 7x +2 $ is irreducible.

I want to show that $f(x) = x^3 + 7x +2 \in \mathbb Z[x]$ is irreducible. My idea is: Let $a$ be a root of it then $2 = -a^3 - 7a$ then $a$ is a divisor of $2$ in $\mathbb Z$ therefore candidates for $...
user avatar
-1 votes
1 answer
73 views

The set of all polynomials whose coefficient of $x^2$ is a multiple of 3.

Here is the question I am trying to know its correct answer: Let $I \subset \mathbb Z[x]$ be the set of all polynomials whose coefficient of $x^2$ is a multiple of 3. Is $I$ an ideal of $\mathbb Z[x]$?...
Emptymind's user avatar
  • 2,069
2 votes
0 answers
94 views

Is $\mathbb Z[x^2]$ an ideal of $\mathbb Z[x]$?

Here is the. Question I am wondering about: Is $\mathbb Z[x^2]$ an ideal of $\mathbb Z[x]$? My question is: My professor defined $\mathbb Z[x^2]$ as the polynomials in which only even powers of $x^2$ ...
user avatar
0 votes
0 answers
19 views

How do we show $F_2[x]/(x^2 + x) ≅ F_2[x]/(x) × F_2[x]/(x + 1) ≅ F_2 × F_2$?

In the polynomial ring section of the textbook that I am reading, it states: $F_2[x]/(x^2 + x) ≅ F_2[x]/(x) × F_2[x]/(x + 1) ≅ F_2 × F_2$. I know how to show the first pair of isomorphism. Since $(x, ...
196884 is 196883 plus 1's user avatar
0 votes
1 answer
37 views

How to prove that the order of $F[x]/(g(x))$ is $|F|^{\deg(g)}$, where $F$ is a finite field?

In the textbook that I am currently learning from, it says that: Given $F[x]/g(x)$ for some fixed $g(x) \in F[x]$ of degree $d ≥ 1$, $$F[x]/g(x) = {r(x) + g(x):\ \deg(r) ≤ d − 1}$$ If $F$ is finite, $...
196884 is 196883 plus 1's user avatar
0 votes
1 answer
55 views

Factorization of $x^{rn} - 1$

Let $q$ be a prime power and $r$ a divisor of $q-1$. Let $n = \frac{q^m-1}{r}$ and $N = rn = q^m-1$. Denote by $\beta$ a primitive element of $\mathrm{GF}(q^m)$ and $\lambda = \beta^{n}$, then $\...
soda's user avatar
  • 33
1 vote
0 answers
101 views

The cardinality of the affine algebraic variety $V_{\mathbb{C}}(I)$ and the dimension of $\mathbb{R}[\mathbf{x}]/I$

I am reading Laurent's notes Sums of squares, moment matrices and optimization over polynomials and, since I am not really experienced in algebraic geometry, I have some questions. This one concerns ...
math_cpt's user avatar
  • 184
0 votes
2 answers
162 views

Prove there is a polynomial $d(x) \in \mathbb Q[x]$ that is a gcd of $f(x)$ and $g(x)$ and whose term of minimal degree is $d_rx^r.$ (D&F #9.3.5(a)).

Here is the question I am trying to understand its solution: Let $R = \mathbb Z + x \mathbb Q[x] \subset \mathbb Q[x]$ be the set of polynomials in $x$ with rational coefficients whose constant is an ...
Intuition's user avatar
  • 3,161
0 votes
1 answer
43 views

Proof of the formal derivative fraction formula [closed]

In a polynomial ring $R[x]$, the “formal derivative” seems to have two formulas, the classic one: $$ f’(x) = a_1 + 2 \cdot a_2 \cdot x + 3 \cdot a_3 \cdot x^2 + \cdots $$ (if $f(x) = a_0 + a_1x + a_2x^...
SchoolStop's user avatar
0 votes
1 answer
56 views

Is there a name for this sort of ideal?

Let $k[x_1,...,x_n]$ be a polynomial ring over a field $k$ and let $R$ be an equivalence relation on $\{1,...,n\}$. Is there a name for the ideal $(x_i-x_j \mid iRj)$?
smoneh's user avatar
  • 11
0 votes
1 answer
90 views

The Ideal given by the kernel of map $ f \in R \mapsto f(0,0) \in \mathbb C[X,Y]$ is not finitely generated for this particular polynomial subring

In the following problem, $R$-module means left $R$-module and $R$ is a ring. I have already proven these facts that may or not be needed: -Show that an $R$-module $ M$ is finitely generated if and ...
some_math_guy's user avatar
2 votes
2 answers
37 views

Can the implication $(x_1 = 0) \rightarrow (p(x_1,...,x_n) = 0)$ be encoded in a system of polynomial constraints in $\mathbb{C}[x_1,...,x_n]$?

Consider a set $S$ of polynomials in $\mathbb{C}[x_1,x_2,...,x_n]$, the polynomial ring of $n$ variables over the complex numbers. The set $S$ can then be interpreted as a system of constraints on the ...
PPenguin's user avatar
  • 930
3 votes
0 answers
75 views

Groebner basis over rational vs finite field

Some algorithms for calculating a Groebner basis are optimized for calculating with coefficients in a finite field. Having determined the basis over a finite field, I'd like to understand what ...
PPenguin's user avatar
  • 930
0 votes
1 answer
81 views

Expressing a multivariate polynomial in terms of its roots

Let $R$ be a commutative ring with unity. So I was given a polynomial $f(x_1,\dots,x_n)$ in $R[x_1,x_2,\dots,x_n]$. I was given the root of $f$ as $(a_1,\dots,a_n)$. My claim was we can write $f(x_1,\...
Caratheodory_Enthusiast's user avatar
0 votes
2 answers
70 views

Number of homomorphisms from $Q[x]/\langle f(x)\rangle$ to $\mathbb{C}$

How many homomorphisms are there from $Q[x]/\langle f(x)\rangle$ to $\mathbb{C}$ that take $1$ to $1$ for an arbitrary polynomial $f(x)\in Q[x]$? I took some examples and tried to figure out the ...
nkh99's user avatar
  • 471
3 votes
1 answer
65 views

Can quotients by polynomials of different degrees be isomorphic as rings?

Let $R$ be an integral domain and let $p(x), q(x) \in R[x]$ be polynomials of different degrees. It is clear that $R[x]/(p(x))$ and $R[x]/(q(x))$ are not isomorphic as $R$-modules because the ...
user02468's user avatar
  • 145
1 vote
0 answers
41 views

Ideal of $F[X_1,\dotsc,X_n]$ generated by polynomials of bounded degree - small generating subset?

Let $F$ be a field, let $S$ be a subset of the polynomial ring $F[X_1,\dotsc,X_n]$, where each polynomial in $S$ has degree at most $d$, and let $I$ be the ideal generated by $S$ (the degree of a ...
Object's user avatar
  • 339
2 votes
1 answer
47 views

Equality of ideals for every value of one variable implies they are equal?

Let $J\subseteq I$ be ideals in a polynomial ring $R=\mathbb{F}[x_1, \ldots, x_n, t]$ over a field $\mathbb{F}$ of characteristic zero. Let's write $I_{\lambda}$ for the ideal in $S=\mathbb{F}[x_1, \...
Simon's user avatar
  • 53
0 votes
0 answers
95 views

Is $\sqrt{-5}$ the only prime in $\mathbb Z [\sqrt{-5}]$ [duplicate]

I am aware that $\sqrt{-5}$ is both prime and irreducible in $\mathbb Z [\sqrt{-5}]$ . And that $2, 3 $ etc., are irreducible but not prime. My question is are there other prime elements in $\mathbb ...
Phalaksha C G's user avatar
0 votes
1 answer
62 views

Infinite intersection of maximal ideals in two variable polynomial ring

Consider the polynomial ring $\mathbf{C}[x,y]$ in two variables. It is a standard fact that this ring is Jacobson and hence the intersection of all its maximal ideals is zero. I am interested to know ...
Jason V's user avatar
  • 352
4 votes
1 answer
73 views

Show that any ideal in $\mathbb{Z}[\sqrt{-5}]$ is generated by only two elements

My first instinct is to use the isomorphism $$\mathbb{Z}[\sqrt{-5}] \cong \mathbb{Z}[X] / (x^2-5)$$ and show that any ideal I in this ring is generated by only two elements. As $\mathbb{Z}[X]$ is ...
Zedssad's user avatar
  • 714
0 votes
1 answer
46 views

Applying ring homomorphism to coefficients of polynomial is a ring homomorphism? [closed]

Let $R_1$ and $R_2$ be rings, $\phi:R_1\to R_2$ some ring homomorphism. Consider the map $\widehat\phi$ that sends any polynomial $f(x) = \sum_ia_ix^i\in R_1[x]$ to $\widehat\phi(f(x)) = \sum_i\phi(...
node196884's user avatar
1 vote
2 answers
117 views

Prime & Maximal Ideal in $\mathbb{Z}[x]$

Question: Is $<x^2+x+1>$ a prime and\or maximal ideal of $\mathbb{Z}[x]$ Background: Relatively new to abstract algebra, hence, would appreciate beginner level answers. My Solution: For Prime: ...
Madhav10612's user avatar
1 vote
0 answers
55 views

Minimal field of definition of the radical ideal $\sqrt{I}$

Let $K$ be a number field, and let $I$ be an ideal of the polynomial ring $K[x_1,\dotsc,x_n]$. Assume that $K$ is the minimal field of definition of $I$. Let $L$ be the minimal field of definition of $...
Object's user avatar
  • 339
2 votes
2 answers
167 views

Isomorphism Quotient Polynomial Rings 2 Var

Examine Whether: $$ \frac{\mathbb{R}[x,y]}{<x^2-y^2-1>} \cong \frac{\mathbb{R} [x,y]}{<xy -1>} $$ Background: 2nd year math undergrad Currently doing introductory ring theory What I Know\...
Madhav10612's user avatar
0 votes
1 answer
81 views

Composition Series and Composition Factors.

I've been studying composition series. I've been struggling with problems related to finding composition series and composition factors. I feel like there have been very limited examples in a lot of ...
HillyBilly's user avatar
5 votes
1 answer
225 views

Orders of all the elements in polynomial quotient ring

Consider a quotient ring $\dfrac{F_p[x]}{(f)}$, where $p$ is prime. I want to find all the possible orders in this ring. I know that with given factorization of $f(x) = f_1(x)^{k_1} * ... * f_n(x)^{...
SarkoxedaF's user avatar
3 votes
1 answer
92 views

Confused on Proof that Polynomials of Degree $n$ has at most $n$ zeroes (counting multiplicity) from Gallian's Contemporary Abstract Algebra

I am a bit confused on the proof that Gallian delivers of the theorem mentioned in the title. Although I have seen other proofs of this fact and have been able to parse them, his specifically trips me ...
adam dhalla's user avatar

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