# 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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\...
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### 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 ...