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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5
votes
1answer
159 views

Is $(x^2+y^2-1, z^2+w^2-1)$ a prime ideal in $\mathbb Q[x,y,z,w]$?

$\newcommand{\Q}{\mathbb Q}$ I saw an argument that the ideal $I=(x^2+y^2-1, z^2+w^2-1)$ is a prime ideal in $\Q[x,y,z,w]$ but I cannot see why. I tried to find a surjective homomorphism from $\Q[x,y,...
4
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0answers
28 views

Function $(f^n)_i:=\begin{cases}1\text{ if }i=n,\\0\text{ else }\end{cases}$ on polynomial ring $R[x]$

Let $R$ be a ring with $1$ and $R[x]$ be the polynomial ring in $x$ over $R$ with pointwise defined addition and convolution as multiplication. Let $f\in R[x]$ be $$f_i:=\begin{cases}1\text{ if }i=1,\...
4
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0answers
266 views

Gauss' lemma for arbitrary integral domains

One of the versions of the classical Gauss' lemma in abstract algebra states the following: Theorem: Let $R$ be an integral domain, $f\in R[X]$ of positive degree and $K$ the quotient field of $R$. ...
3
votes
1answer
95 views

Number of maximal ideals of $F_q[x_1,…,x_n]$

I am currently studying commutative algebra and came across the following question. Let $F$ be a finite field with $q$ elements, let $A=F[x_1,...,x_n]$ and denote by $m$ a maximal ideal in $A$. ...
3
votes
1answer
51 views

Ideals in polynomial ring

I am facing this problem: Prove that there is a bijection between the monic divisors of $x^n−1$ in $F[x]$ and the ideals of $F[x]/\left<x^n−1\right>$. I tried to find how the ideals in $F[x]/...
3
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0answers
75 views

determining if quotient ring of polynomials over a finite field is a field or not

I am stuck with this question: "Determine if $GF(2011^2) [x] /<x^4-6x+12>$ is a field or not." I know that since this polynomial ring is defined over a field, I only have to determine if $x^4-...
3
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0answers
48 views

Algorithm to find relations between polynomials

Let $p_1,\ldots,p_k\in\mathbb{C}[x_1,\ldots,x_n]$ be polynomials. Is there an algorithm to compute the ideal of relations between them? More precisely, to find a set of generators for the kernel of ...
3
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0answers
34 views

Prime ideals in polynomial rings satisfying a particular condition

In the paper "Kronecker function rings and flat $D[X]$-modules" by Arnold and Brewer it's proved the following result: Lemma: Let $D$ be an integral domain. If $Q$ is a prime ideal of $D[X]$ such ...
3
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0answers
594 views

Polynomial rings over a commutative ring with identity

Let $R$ be a commutative ring with identity. Consider the polynomial ring in $n$ indeterminates $R[X_{1},\ldots ,X_{n}]$. Let $R[Y_{1},\ldots ,Y_{m}]$ be a polynomial sub-ring of it in $m$ ...
2
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0answers
56 views

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 ...
2
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0answers
39 views

Cyclic multiplicative subgroups

Let $R[t]$ be the polynomial ring over the nonzero commutative ring $R$ and $f_R$ be the associated polynomial function. If $|\{\alpha \in R : f_R(\alpha)=0\}| \leq \deg(f)$ for every $0 \neq f \...
2
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0answers
51 views

Classifying homomorphisms on polynomial rings with real coefficients.

Show that every homomorphism $\mathbb{R}$[X] $\rightarrow$ $\mathbb{R}$[X] can is equal to $φ_g$ for a unique g $\in$ $\mathbb{R}$[X], given by $φ_g(f)$ = $f(g(X))$ My guess for any homomorphism $h$, ...
2
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0answers
69 views

Prove that any element of this group ring is an associate to an element of the polynomial ring and that the group ring is an integral domain

Let $G = \langle x\rangle$ be the infinite cyclic group generated by an element $x$. The group ring $R = Q(G)$ consists of finite sums of the form $r_{−m}x^{−m} + r_{−m+1}x^{−m+1} + · · · + r_{−1}x^{−...
2
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0answers
49 views

A question about field of fractions of a ring of functions

Let $k$ be a field, $f(x,y) \in \bar{k}[x,y]$ an irreducible polynomial. Define $C_f := \bar{k}[x,y]/(f)$ and $\bar{k}(Z_f) = Frac(C_f)$. Now, there is an equality and a claim that I can not ...
2
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0answers
56 views

Relatively simple algorithm for factoring polynomials in $GF(2^k)[x]$ with $32 \leq k \leq 128$

Is there a relatively simple algorithm for factoring polynomials in $GF(2^k)[x]$ with $32 \leq k \leq 128$? I like McEliece's algorithm for factoring polynomials in $GF(2)[x]$, because of its ...
2
votes
0answers
137 views

Formal derivative of polynomials with coefficients in a quotient ring $GF(2)[x]/p(x)$

How do you determine the formal derivative of polynomials with coefficients in a quotient ring $GF(2)[x]/p(x)$? I am comfortable with calculating formal derivatives of polynomials with coefficients ...
1
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0answers
15 views

Eigenvalues of an endomorphism over a polynomial ring

I am currently preparing for a math exam and am stuck on the following question: Let $\Bbb K$ be a field, let $\Bbb K[T]$ be the polynomial ring over the variable $T$ over $\Bbb K$, and let $\...
1
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0answers
32 views

Confusion about length of module over different rings and Macaulay2 code about computing length of module.

I was trying to compute some examples dealing with length of modules and got stuck with this simple example: Let $R=k[t]/(t^2)$ where $k$ is a field and $J=(x^3)$ be the ideal of the polynomial ring $...
1
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0answers
40 views

Krull dimension of polynomial ring in one variable

I am currently working my way through Bosch, Algebraic Geometry and Commutative Algebra. I want to solve Exercise 1, Chapter 2.4: Consider the polynomial ring $R[X]$ in one variable over a not ...
1
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0answers
33 views

Find an ideal $I$ in $A$ so that $A/I$ is a finite field of $25$ elements.

Let $A = \frac {\Bbb Z[X]} {\left ( X^4+X^2+1 \right )}.$ Find an ideal $I$ in $A$ such that $A/I$ is a finite field of $25$ elements. I have seen that the polynomial $X^4+X^2+1$ is reducible in $\...
1
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0answers
24 views

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 ...
1
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0answers
49 views

$\mathbb{ R } \left[ x _ { 1 } , \ldots , x _ { n } \right] / \left( x _ {1} ^ {2} + \cdots + x _ {n} ^ {2} - 1 \right)$ is a UFD when $n>2$

Prove $$\mathbb { R } \left[ x _ { 1 } , \ldots , x _ { n } \right] / \left( x _ { 1 } ^ { 2 } + \cdots + x _ { n } ^ { 2 } - 1 \right)$$ (i) is not a UFD when $n=2$ (ii) is a UFD when $n&...
1
vote
0answers
20 views

Answer verification to question about polynomial rings and derivatives.

Can someone check whether my answers are okay to the following Judson question? (a) Let $f(x)=a_{n}x^{n}+\cdots +a_{m}x^{m}+\cdots a_{1}x+a_{0}$ and $g(x)=b_{m}x^{m}+\cdots +b_{1}x+b_{0}$. Assume ...
1
vote
0answers
93 views

Krull-Dimension of $k[X, Y , Z]/(Y-Z^2, XZ-Y^2)$

I am supposed to compute the krull-dimension of $k[X, Y, Z]/(Y-Z^2, XZ-Y^2)$, where $k$ is a field, and feel quite at sea, as we have only looked at principal ideals and obvious applications of the ...
1
vote
0answers
99 views

Constructing reducible polynomial with two irreducible polynomial

I'm currently trying to prove following statement. Let f,g be homogeneous polynomials of degree n,m respectively, in $k[X,Y,Z]$. Here k is algebraically closed field, and n $\le$ m. Also f does not ...
0
votes
0answers
31 views

Description of ideals of a polynomial ring

$R$ is an integral domain then shows that an ideal $I$ of $R[X]$ is either principal or of the form $(f, r), r \in R$. The hint is to use the Gauss lemma. I started by assuming that $I$ is not ...
0
votes
0answers
16 views

How can I simplify $\mathbb{F}[X,Y]/(X^2-Y^2)$?

Here $\mathbb{F}$ is an arbitrary field. $$\psi :\mathbb{F}[X,Y]\rightarrow\mathbb{F}[X]\times\mathbb{F}[X];\\ f\mapsto (f \ \text{mod} (X+Y),f\ \text{mod} (X-Y))$$Is a ring homomorphism with kernel $...
0
votes
0answers
35 views

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....
0
votes
0answers
7 views

Hilbert polynomials in two variables with Macaulay2

In J. Symb. Comput. (1999) 28, 681-710, Levin worked with bifiltered, finitely generated $R$-modules ($R$ being a polynomial ring in two sets of variables) and he found an analogue of the Hilbert ...
0
votes
0answers
48 views

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 $k$, ...
0
votes
0answers
33 views

Field of fractions and rational functions

When I have a field $k$ and take the ring of polynomials in the variables $x1, x2, ..., xn$, and subsequently take the quotient field of these polynomials, I was asking myself, is this in 1:1 ...
0
votes
0answers
38 views

$\gcd(x,y)=1$ then $\gcd(c \cdot x, c \cdot y)=c$ for polynomials

As a part of a proof I want to use that for polynomials $c, x,y$ it is the case that $\gcd(x,y)=1$ then $\gcd(c \cdot x, c \cdot y)=c$. Is this always true?
0
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0answers
45 views

Trouble with interpreting a question about polynomial rings?

I trying to do problem $9.3.1$ in Dummut & Foote; I think they use some "abuse of notation" that I don't understand. Let $R$ be an integral domain with quotient field $F$ and let $p(x)$ be a ...
0
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0answers
47 views

Relationship between $F(x, y)$ and $F[x, x^2y, x^3y^2, …, x^{n+1}y^n, …]$, is my understanding correct?

I am doing problem $8$ in section $9.1$ in Dummit and Foote: Let $F$ be a field and let $R = F[x, x^2y, x^3y^2, ..., x^{n+1}y^n, ...]$ be a subring of the polynomial ring $F[x, y]$. $a)$ ...
0
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0answers
30 views

Help with fixing proof that transcendental extension of $\mathbb{Q}$ is dense in transcendental extension of $\mathbb{Q}_{p}$

Good morning. I am trying to prove the following statement (I suspect it is true): Consider a finite set $(a_{1},...,a_{n})$ of elements in an elementary extension $*\mathbb{Q}_{p}$ of $\mathbb{Q}_{p}...
0
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0answers
58 views

Is the cardinality of the polynomial quotient ring $\mathbb{Z}_n [x] /f(x)$ always finite?

For some polynomial $f(x)$ does the polynomial quotient ring $\mathbb{Z}_n [x] / f(x)$ always have finite cardinality? Also, I have noticed that the examples of polynomial (over $\mathbb{Z}_n$) ...
0
votes
0answers
59 views

Polynomial differentiation map

I’m studying rings of polynomials And I don’t understand the following exercise. “Let F be a field and char(F)=0 Let D : F[x] -> F[x] defined by D(f(x))=f’(x) Find image of F[x] under D” The ...
0
votes
1answer
27 views

How to show each element of $\frac{Q[x]}{I}$ is of the form $a_0+a_1t+a_2t^2$

Consider the polynomial ring $Q[x]$. Let $p(x) = x^3-2$. Let $I$ be an ideal generated by $p(x)$. Show that each element of $\frac{Q[x]}{I}$ is of the form $a_0+a_1t+a_2t^2$ with $a_0, a_1, a_2$in $Q$ ...
0
votes
1answer
203 views

Find the sum and product of $f(x)=3x-5, g(x)=2x^2-4x+3$ in $Z_8$.

Find the sum and product of $f(x)=3x-5, g(x)=2x^2-4x+3$ in $Z_8$. Does using high school style of solving this yields a different answer? What would be the answer?
-1
votes
0answers
20 views

Show that $R[x_1,\dots x_n]/IR[x_1,\dots,x_n]\cong(R/I)[x_1,\dots,x_n]$

To simplify the notation let $X:=x_1,\dots,x_n$. Let $R$ be a commutative ring with a unit and let $I\subseteq R$ be an ideal. Let $IR[X]$ be the ideal in $R[X]$ generated by $I$. Build an ...