# 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".

136 questions
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### Zeros in Splitting Factor Rings

I am looking at the following proof that $x^3-x-1$ splits in an extension field of $F_{3}[x]$. Let us consider the field $$K = F_3[x]/\langle x^3-x-1\rangle$$ If $\theta$ is the image of $x$ in $K$, ...
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### Proving $f(x)=x^4+4x^3+3x^2+7x-4$ is irreducible

Here's my attempt, I'm almost there but I'm stuck: Using a hint, I wrote the modular reduction: Reducing the coefficients modulo $2$ gives: $\left [ f \right ]_2=x^4+x^2+x=x(x^3+x+1)$. Reducing ...
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### Polynomial rings and ideals

I'm trying to learn some algebra by myself and I need some help. $I=(x^2,x^3)$ an ideal in $R[X]$. Give an example of two polynomials with exactly four terms, one that is in $I$ and one that isn't. ...
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### 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 ...
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### Clarification about a proof regarding a sum of polynomials being expressed as a linear combination of S-polynomials

I'm reading this proof from Ideals, Varieties, and Algorithms by David A. Cox, Donal O'Shea, and John Little. You can find an online version here. This is Lemma 5 of Chapter 2, page 85. From my ...
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### Difference between $F[x,y]$ and $F(x)[y]$ in Ring theory

I come across following ring $F[x,y]$ and $F(x)[y]$ where $F$ is a field. I think both are same initially . But as $xy$ is irreducible in $F(x)[y]$ but reducible in $F[x,y]$. Which is a ...
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### Surjections of polynomial rings, Krull dimension, and regular sequences

Let $k$ be a commutative Noetherian ring with unity (I will ultimately want $k = \mathbb Z$) and $$\varphi\colon A = k[x_1,\ldots,x_n] \to k[y_1,\ldots,y_p] = B$$ a surjective $k$-algebra homomorphism....
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### Prime ideals in multi-variable polynomial rings over $\mathbb{Z}$

There is a nice classification of prime ideals in the ring $\mathbb{Z}[x]$, see this question. Is there any generalization of this result, on $\mathbb{Z}[x_1,\cdots,x_n]$? Due to this post, I ...
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### 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, ...
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### 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 ...
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### 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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### 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 ...
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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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### 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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### 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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### 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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### How the following multiplication table is solved ( related to $F_2[X]/f(x)$ ) [duplicate]

$F_2$ 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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### 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 $G(x, y)=G(y,x)$.