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 Polynomial Function on a Circle that Vanish at a Point

Let $R = \mathbb{R}[x,y]/(x^2+y^2-25)$ and $I$ the ideal of functions which vanish at the point $P = (3,4)$. I have proven that $I$ is generated by $(x-3,y-4)$ and that if $I= (f)$ for some $f \in R$, ...
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$\phi: \mathbb{R}[X] \to \mathbb{C}$ is a homomorphism such that: $\phi(X) = 1 + i$. What is $\ker \phi$?

$\phi: \mathbb{R}[X] \to \mathbb{C}$ is a homomorphism such that: $\phi(X) = 1 + i$. What is $\ker \phi$? In my thinking $\ker \phi = \{0\}$ because there is no way to add or multiply $1 + i$ in such ...
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$\mathbb{C}[x,y,z]/(x^2+y^2+z^2-1)$ is not a UFD

Wiki says that the coordinate ring $\mathbb{C}[x,y,z]/(x^2+y^2+z^2−1)$ of the complex sphere is not a unique factorization domain. I want to know why it is not a UFD. We denote $X,Y,Z$ the residue ...
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Show that $(x^3+x+1)$ is prime in $\mathbb{Z}/(2)[x]$ [duplicate]

I want to show that the ideal $(x^3+x+1)$ is prime in $\mathbb{Z}/(2)[x]$. I know that $\mathbb{Z}/(2)[x]$ is isomorphic with $\mathbb{Z}[x]/(2)$ and also know that since $\mathbb{Z}/(2)[x]$ is ...
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Determining if two ideals in $\mathbb{Q}[X,Y]$ are equal

I'm looking through old algebra exercises and came across this: In $\mathbb{Q}[X,Y]$ let $$G=(X-1,Y), H=(XY+X-1,2X-Y-2)$$ and $$I= \langle G\rangle, J=\langle H \rangle$$ I want to determine if $I=J$...
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if every finitely generated $R$-module has Finite Free Resolution, then every finitely generated $R[x]$-module also has it

I am reading the Rotman's book "An Introduction to Homological Algebra" and I do not understand the part of the proof of theorem 8.47 in page 481. Theorem 8.47 Let R be a commutative ...
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T[x] Euclidean if and only if T is a field. [duplicate]

I am for the first time teaching Algebra 1 practicals. With students, we were proving that If T is a field, then the ring of polynomials T[x] is Euclidean Now I ...
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If a $k$-algebra $R$ is isomorphic to $k[x_1,\dots,x_n]/J$ (where $J$ is an ideal), then $R$ is finitely generated.

I have a question for the following argument. My goal is to find a finite set of elements of $R$ that generate it. To that end, I let $\phi$ be the isomorphism from $k[x_1,\dots,x_n]/J$ to $R$, and I ...
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Faulty proof in the claim "If a polynomial is irreducible in $Z_p [x]$ then it is irreducible in $Z_{p^k} [x]$?

I was reading a paper regarding the divisibility of polynomials modulo a composite number. The paper can be found here: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.821.1301&rep=rep1&...
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Isomorphism from a non trivial ring onto the polynomial ring over itself [duplicate]

Does there exist a non-trivial ring $R$ such that $R\cong R[X]$? ($R[X]$ denotes the polynomial ring over $R$). If not, how does one prove it? Else, an example of such a ring and the respective ...
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A not constant unit in the ring $\mathbb{Z}/27\mathbb{Z}[X]$ and another one in $\mathbb{Z}/96\mathbb{Z}[X]$.

Could someone help me to find a unit (not constant) in the ring $\mathbb{Z}/27\mathbb{Z}[X]$ ? And another one in the ring $\mathbb{Z}/96\mathbb{Z}[X]$ ? I have done the following: Let we take a ...
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Understanding the structure of $\mathbb{Z}[[x]]/(x-x^2)$

I'm currently trying to understand the structure of quotients of power series rings, and found a particular example I'm confused about. Let $f = x-x^2$ be a polynomial in $\mathbb{Z}[[x]]$, and ...
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Kernel of the polynomial to function map?

It's common to think of polynomials as functions, but it was recently brought to my attention that this isn't exactly right. Consider, for example $x$ and $x^7$ in $\mathbb{Z} / 7\mathbb{Z} [x]$. ...
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Showing $k[X] \cong k[X,Y,Z]\big/{(Y-X^2,Z-X^3)}$

Let $k$ be an algebraically closed field. I want to show that $(Y-X^2,Z-X^3)$ is a prime ideal of $k[X,Y,Z]$. I know that an ideal $\frak a \subseteq$ $R$ of a (commutative ring with $1$) is prime if ...
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Show that $R[x]/\langle P \rangle$ is a free R-Module and R-Algebra

Show that $I=R[x]/\langle P \rangle$ is a free R-Module and finite generated R-Algebra for $P\in R[x]$ a polynomial of degree d, R an integral domain and $P$ normed(scaled?). What would happen if it ...
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Is $\mathbb{C}[X, Y] / (X^5 + Y - 13)$ integrally closed?

I want to check if $R := \mathbb{C}[X, Y] / (X^5 + Y - 13)$ is a Dedekind domain or not. I know $R$ is an integral domain because $X^5 + Y - 13$ is prime in $\mathbb{C}[X, Y]$. $R$ is also noetherian ...
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Prove that $\mathbb{F}_2[X] / \left(x^3 + x + 1 \right)$ is a field with 8 elements

Just want to know if my proof is correct. First of all, it is easy to check that $f(x)=x^3+x+1$ is irreducible over $\mathbb{F}_2$. This implies that $\frac{\mathbb{F}_2[X]}{(f(x))}$ is indeed a field....
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Why a field $F$ sits naturally inside the ring $F[x]_{m(x)}$ for $m(x)\in F[x]$?

I am reading exercise page 231 from Ronald S. Irving's book: Integers, Polynomials and Rings. I can understand and solve problem in part 1. I get ring $\mathbb{F}_2[x]_{x^2}$ is not a field. But, I am ...
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Classification of commutative rings $R$ satisfying $\dim(R[T])=\dim(R)+1$

Let $R$ be a commutative ring. Then $\dim(R[T]) \geq \dim(R)+1$. Is there a classification of those commutative rings with the property $\dim(R[T])=\dim(R)+1$? Every Noetherian commutative ring has ...
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Showing $\langle x,y \rangle$ is a prime ideal of $K[x,y]$

The biggest thing that it stopping me from progressing in this question is the fact that I have two elements in the ideal, this topic it still very new to me. I can show that $I=\langle x \rangle$ and ...
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A confusion about showing $g$ is a unit in which ring.

Proving that $(ac - (b^2 + 1))$ is irreducible in $k[a,b,c].$ Here is my trial: I usually work by degrees and taking the lower degree of my polynomial, but the thing here is that I have two variables ...
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Let be $R$ a $\mathbb{C}-$ algebra and take $R\otimes_{\mathbb{C}}\mathbb{C}[[t]]$, Is this isomorphic as ring to $R[[t]]?$ I think that I can prove it using the quozient of these ring and take the ...
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How exactly are polynomials related to their evaluation maps?

My abstract algebra textbook* has taken great care in the last few sections to stress that, "polynomial are NOT functions to just 'plug' values of x into". To fully 'get' this, and to 'get' ...
$k[x,y,z]$ and modules.
My professor gave us this example on modules (he started this by saying what is a basis? what is the meaning of linearly independent?): $R = k[x,y,z]$ where $k$ is a field and $I = xyR + yz R + xz R$....