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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61 views

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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1answer
56 views

$\phi: \mathbb{R}[X] \to \mathbb{C}$ 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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2answers
173 views

$\mathbb{C}[x,y,z]/(x^2+y^2+z^2-1)$ 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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25 views

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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3answers
56 views

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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39 views

Let $D$ be an integral domain and let $c\in D$ be irreducible in $D$. Show the ideal $(x,c)$ in $D[x]$ is not principal. [duplicate]

Question: Let $D$ be an integral domain and let $c\in D$ be irreducible in $D$. Show the ideal $(x,c)$ in $D[x]$ is not principal. Thoughts: Since $c$ is irreducible in $D$, $c$ is noninvertible in $...
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0answers
21 views

Rational functions: unique representation as a polynomial fraction?

Let $\mathbb{F}$ be a field. And let $\mathbb{F}(\bar x)$ be the field of rational functions over the field $\mathbb{F}$ in the indeterminate $x_1,\dots,x_n$. Namely, all the functions $f:\mathbb{F}^n\...
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0answers
97 views

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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1answer
39 views

Is $\Bbb Z_4 [X]$ an integral domain?

Is $\Bbb Z_4 [X]$ an integral domain? Explain why you cannot use the fact that if a commutative ring $R$ is an integral domain, then $R[X]$ is an integral domain. If $P,Q \in \Bbb Z_4[X]$, then $\...
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2answers
53 views

Issues with ideals

We're trying to understand the field $\mathbb{Z}_2[x]/\langle x^3+x^2+1 \rangle$, and in particular why it has eight elements. I know that $x^3+x^2+1$ is irreducible in $\mathbb{Z}_2$ and so factor is ...
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1answer
18 views

Domain containing polynomial ring over field, if finitely generated generated as module over it, is free as module over it

This is the first part of exercise 4.2 from Eisenbud's "Commutative Algebra" textbook. The problem is: Let $R$ be a domain containing a polynomial ring in one variable over a field, say $R \...
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0answers
19 views

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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31 views

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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1answer
32 views

How to understand $I(\emptyset) = k[x_1,\dots,x_n]$?

In our class, for $Z \subseteq \mathbb{A}^n$, we define: $$I(Z) := \{f \in k[x_1,\dots,x_n] \ | f(z) = 0 \ \forall z \in Z\}$$ I understand this as saying that, given a set of points $Z$ in $\mathbb{A}...
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1answer
51 views

Mapping from $F$ to $E$ where $F$ is a field and $E=F[x]/\langle p(x)\rangle$

Here is the question as given that I am having trouble with: Let $F$ be a field and $p(x)$ an irreducible polynomial in $F[x]$. In this investigation we showed that $E=F[x]/\langle p(x)\rangle$ is a ...
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1answer
29 views

Let $J \subset k[x_1,\dots,x_n], V(J) = Z \in \mathbb{A}_k^n$, and $A(Z) := k[x_1,\dots,x_n] /J$. Then $f \in A(Z)$ becomes a function $f: Z \to k^1$.

Here $V(J)$ is the vanishing locus of $J \subset k[x_1,\dots,x_n]$. The above is a remark from my lecture that I can’t wrap my head around. If $f \in A(Z)$, shouldn’t $f$ be a coset of $J$ in $k[x_1,\...
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1answer
47 views

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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0answers
55 views

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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1answer
44 views

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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1answer
40 views

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 ...
3
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1answer
52 views

Sutructure of the ideal of vanishing polynomials on $\mathbb{F}_p$

Let $\mathbb{F}_p$ denote the field with $p$ elements. I will write $\mathbf{x}$ for $(x_1,\dots,x_n)$. We're interested in the structure of the ideal $I=\{f\in\mathbb{F}_p[x_1,\dots,x_n]\mid \text{ev}...
4
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2answers
97 views

"Mod" out symmetry in ideal for a Groebner basis calculation (using a quotient ring?)

Consider a set of polynomials $P$ in the polynomial ring $R$ of $n$ variables ($R = \mathbb{C}[x_1,...,x_n]$), and let $I$ be the ideal generated by the polynomials in $P$. I have an ideal which is ...
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1answer
42 views

Let $g(x),h(x)\in\mathbb{Z}[x]$ and $h(x)$ is monic. If $h(x)$ divides $g(x)$ in $\mathbb{Q}[x]$ then $h(x)$ divides $g(x)$ in $\mathbb{Z}[x]$

Question: Let $g(x),h(x)\in\mathbb{Z}[x]$ and $h(x)$ is monic. If $h(x)$ divides $g(x)$ in $\mathbb{Q}[x]$ then $h(x)$ divides $g(x)$ in $\mathbb{Z}[x]$ I know that, as $h(x)$ divides $g(x)$ in $\...
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0answers
43 views

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 $\mathbb{Z} / 7\mathbb{Z} [x]$. ...
3
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1answer
76 views

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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1answer
50 views

Proof that $(v_n)=(1+t)^n$ is a basis for $K[t]$

Prove that $(v_n)=(1+t)^n$ is a Basis for $K[t]$. I cant figure out how to prove this statement. I need to show linear independence. I tried two things: The first one is with induction: Let $$0=\...
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1answer
62 views

Simplifying a tensor product

Problem: Show that $$\mathbb{C}[X,Y]/\langle{XY-1}\rangle\otimes_\mathbb{C}\mathbb{C}[X,Y]/\langle{X^2-Y}\rangle\cong \mathbb{C}[X,X^{-1},Z],$$ $$\mathbb{C}[X,Y]/\langle{XY-1}\rangle\otimes_{\mathbb{C}...
3
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1answer
68 views

Factorizations and roots of a bivariate polynomial

Problem: Show that for $g(x,y)\in \mathbb{C}[x,y]$ and $t$ an indeterminate, $g(t^2,t^3)=0$ if and only if $g$ is divisible by $y^2-x^3$. Attempt: If $g$ is divisible by $y^2-x^3$, then obviously $g(t^...
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1answer
42 views

How do I go between generators and elements of an ideal in $𝔽_2[x_1,...,x_n]$?

Part 1: Elements of ideals defined by generators (answered) Consider the ideal $I=⟨f_1,f_2,f_3⟩⊂𝔽_2 [x_1,x_2,x_3,x_4 ]$. $f_1,f_2$ and $f_3$ are generators of the ideal $I$. $f_1=(1−x_1)x_2 \\ f_2=...
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0answers
41 views

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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1answer
47 views

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 ...
6
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1answer
86 views

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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1answer
32 views

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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1answer
37 views

Finding the inverse of $(x^2+2)$ in the field $S = \Bbb F_3[x]/(x^4+x^3+x^2+1)$

Let $S = \Bbb F_3[x]/(x^4+x^3+x^2+1)$. Find the inverse of $(x^2+2)$ in $S$. I know I'm looking for a polynomial $q(x)$ such that $(x^2+2)q(x) = 1 \mod x^4+x^3+x^2+1$ i.e $(x^2+2)q(x) + k(x)(x^4+x^3+x^...
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1answer
49 views

Prove: If $f,g \in \Bbb Z[x]$ then $C(fg) = C(f)C(g)$

If $f,g \in \Bbb Z[x]$ then $C(fg) = C(f)C(g)$. C is the content of a polynomial (greatest common divisor of the coefficients). The proof states that proving: "For any prime, $p$ we have $p|C(fg)$...
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1answer
37 views

Show that $K[x]/(x)$ is isomorphic to $K$, where the map is defined by sending a polynomial to a constant coefficient

I have a problem here. show that $K[x]/(x)$ is isomorphic to $K$, where the map is defined by sending a polynomial to a constant coefficient. My attemot on this problem is to defined a function $f: K[...
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0answers
138 views

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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2answers
85 views

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 ...
0
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1answer
59 views

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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0answers
43 views

Exercise about tensor product

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 ...
1
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1answer
53 views

Are there specific properties defining the kernel of a quotient ring?

I'm asking because the correction of an exercise of a course I'm following (commutative algebra) has the following conclusion: Let I be an ideal of a commutative ring A, and $I[X]=\{\sum_{k=0}^{n}i_{...
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0answers
60 views

Which is true for the ring $\dfrac{\mathbb{R}[x,y]}{\langle x^3-y^2\rangle}$

The ring $$\dfrac{\mathbb{R}[x,y]}{\langle x^3-y^2\rangle}$$ a) is unique factorization domain and principal ideal domain b) is unique factorization domain. c) unique factorization domain but not ...
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0answers
24 views

a problem of polynomial over non-commutative ring

Let $R$,$S$ be a ring with identity. If they are commutative rings, we have a useful theorem. theorem:let $φ$ be a homomorphism from $R$ to $S$.Let $s∈S$,then there isba unique homomorphism $ψ$ from $...
3
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1answer
163 views

Ideals of $\mathbb{C}[x,y]$ contains $x^2 + y^2 - 5$ and $xy - 2$?

This is a question from Artin's algebra Chapter 11, 9.8. I am trying to consider the quotient ring $\frac{\mathbb{C}[x,y]}{(x^2 + y^2 - 5, xy - 2)}$ which imposes the relation $x^2 + y^2 - 5 = 0$ and $...
0
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0answers
26 views

Matrix-vector multiplication in $(Z_q[X]/(X^n+1)$ using NTT representation

Let $R_q = Z_q[x]/(X^n+1)$, $n=256$, $q$ large prime be a polynomial ring, and let $A \in R_q^{K \times L}$ be a matrix of polynomials in this ring and $v \in R_q^{L}$ a vector of polynomials. Now we ...
0
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1answer
95 views

Submodules and $T$-invariant subspaces.

My professor wrote this on the board: Claim: there is a bijection between submodules and $T$-invariant subspaces {$T(w) \subset W$}. Proof: If $W \subset V$ is $T$-invariant, then $f(X).w = f(T)(w) \...
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2answers
87 views

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' ...
2
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0answers
58 views

$k[x,y,z]$ 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 $I = xyR + yz R + xz R$....
5
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1answer
622 views

Exercise III.5.10 in Grillet's Abstract Algebra

Let $R$ be commutative, with characteristic either $0$ or greater than $m$. Show that a root $r$ of $A \in R[X]$ has multiplicity $m$ if and only if $A^{(k)}(r)=0$ for all $k<m$ and $A^{m}(r) \neq ...
1
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1answer
47 views

Is $(\Bbb Q[x])_x/(\Bbb Q[x])$ an Artinian $(\Bbb Q[x])$-module?

Consider the ring $R = \Bbb Q[x]$ and the multiplicatively closed set $S = \{1, x, x^2, \ldots\}$. Let $R_x = S^{-1}R$. Consider the module $M = R_x/R$. Determine whether $R_x$ and $M$ are Noetherian/...

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