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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0answers
6 views
Towering of ideals
Is it true that $\mathbb{Z}[x,y]/(p,q) \simeq (\mathbb{Z}[x,y]/(p))/(q)$, where $p \in \mathbb{Z}[x]$ and $q \in \mathbb{Z}[y]$?
0
votes
1answer
23 views
Applying degree 2 or 3 irreducibility tests to higher degree
Given the polynomial$\ x^4+x+1$, I have to find out if it is irreducible over $\mathbb Q $.
When looking at the solutions, they applied the degree 2 or 3 irreducibly tests to determine that it ...
3
votes
2answers
54 views
Find all of homomorphisms $Φ$ from $\mathbb C$[$x$]/$I$ to $\mathbb C$ that satisfies specific condition
I want to know how can I find ALL homomorphism that satisfies the condition mentioned in the question above. Please give me a method or answer in detail.
1
vote
2answers
30 views
GCDs for the polynomial ring over a Galois field.
You can find many examples of computing the inverse of an element inside a Galois field. (For example here)
What happens if we look at the polynomial ring over a Galois field and would like to ...
0
votes
2answers
47 views
coprime ideals in $K[X]$
If $K$ is a field, $A=K[X]$, take $m,n \in K$ such that $m \ne n$. Prove that the ideals $I=(X-m)$ and $J=(X-n)$ are coprime.
I know the regular definition of coprime. But here, should we prove $I + ...
1
vote
1answer
22 views
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 ...
0
votes
0answers
21 views
Heuristics for finding minimal primes/ primary decomp. Example: (XZ-Y^2, X^3-YZ)
In occassions where I need to find the minimal primes associated to an ideal / find a primary decomp., sometimes I can do it just fine, sometimes I find myself completely blind in the search for it.
...
0
votes
0answers
34 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
2answers
72 views
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 ...
1
vote
1answer
38 views
Please help me understand “The Weak Nullstellensatz.”
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 ...
0
votes
1answer
58 views
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 ...
6
votes
1answer
44 views
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 $\mathbb{Q}[x, y]$. ...
1
vote
2answers
44 views
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?
3
votes
1answer
46 views
A computational criterion of irreducibility in $\mathbb Z[X]$?
If $f\in\mathbb Z[X]$ and there are $x_1,\dots, x_n\in\mathbb Z_+$, where $n>\deg f$, such that $f(x_i)\in\mathbb P$, $i=1,\dots n$, then $f$ is irreducible over $\mathbb Z$. Because, if $\,f=g\...
0
votes
0answers
27 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
2answers
66 views
Finding units and zero divisors in a polynomial quotient ring
I am trying to study for an exam and I am not sure of this solution my professor posted to an exercise. I am given the polynomial quotient ring $\mathbb{Z}_6/(x^2+2x)$ and have to find all units and ...
0
votes
2answers
38 views
Prove $(0) = (x)\cap (xz^{n-1} + \lambda y^n)$ in $R=\frac{k[x,y,z]}{(x^2,xy)}$
Studying for my algebra final and doing some practice problems, and I can't seem to understand this one...
Full problem:
Let $k$ be a field, and $R=\frac{k[x,y,z]}{(x^2,xy)}$. For $n\in\mathbb{N}, \...
1
vote
2answers
41 views
How the following multiplication table is solved ( related to $F_2[X]/f(x)$ )
$F_2$ 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 ...
2
votes
0answers
27 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 \...
1
vote
0answers
24 views
Isomorphism between quotient fields of polynomial rings [closed]
Let $R$ be an integral domain and $F$ be its field of fractions. If $X$ is a nonempty set, prove that there is a ring monomorphism from $R[X]$ to $F[X]$ that extends to an isomorphism of their ...
1
vote
1answer
42 views
Homomorphisms and automorphisms on polynomial rings
I am trying to prove a series of propositions:
Given any homomorphism p from $\mathbb{R}$[X] to $\mathbb{R}$[X], show that it is equal to $\phi_g$ for a unique g in $\mathbb{R}$[X], with $\phi_g$(f) =...
2
votes
0answers
27 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 $h$, ...
1
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0answers
17 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 ...
2
votes
1answer
72 views
Fundamental ismorphism theorem
I don't understand how to apply the fundamental isomorphism theorem to polynomial quotient rings. For example is the ring $\mathbb C[X,Y,Z]/\langle X^2-Z,XZ-Y^3\rangle$ isomorphic to $\mathbb C[X,Y]/\...
-1
votes
2answers
54 views
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 $G(x, y)=G(y,x)$.
5
votes
1answer
131 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,...
0
votes
1answer
37 views
If $f|d$, then $f$ is invertible in $R[X]/(d)R[X]$
Let $R$ be a field and $f$ and $d$ be polynomials in $R[X]$. If $f|d$, then $f$ is invertible in $R[X]/(d)R[X]$.
I tried to either prove of disprove this statement, but so far I haven't been able ...
1
vote
1answer
24 views
No element of degree $0$ (constant $= a_0 X^0$) in $\mathbb{Z}_4/(2X) \mathbb{Z}_4$
Consider $\mathbb{Z}_4[X]/(2X) \mathbb{Z}_4[X]$. Then we wish to show/verify that the residue class $X$ does not contain an element of degree $0$.
In the previous exercise the book asked that if we ...
0
votes
2answers
30 views
$a^4=0$ then $1-a$ is invertible in $R[x]/(d)R[x]$ [duplicate]
Suppose $a^4=0$, for some $a \in R[x]/(d)R[x]$, then prove that $1-a$ is invertible.
I was thinking since $a^4 = a \cdot a \cdot a \cdot a=0$, this implies that $a$ has to be zero (?) . Now we have ...
1
vote
1answer
49 views
The maps $f_+$ and $f_-$ are well-defined.
We define the maps $f_+$ and $f_-$ from $\mathbb{Q}[X]/(X^2 -2) \mathbb{Q}[X]$ to $\mathbb{Q} + \mathbb{Q} \cdot \sqrt2$ in the following manner: For any residue class $g+ (X^2-2) \mathbb{Q}[X]$, we ...
1
vote
1answer
42 views
Prove $x^4 +x+1$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[x]$ [duplicate]
$ x^4 +x+1$ in $\mathbb{Z}/2\mathbb{Z}[x]$ is an irreducible polynomial.
So far we have only treated quadratic and cubic polynomials, which are irreducible if they do not have any zeros. However, ...
3
votes
0answers
69 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-...
1
vote
1answer
35 views
What are the polynomial quotients $R/x$ and $R/(R/x)$ for $R = (\mathbb{R}[x]/x^n)$?
Define the polynomial ring quotient $R = \mathbb{R}[x]/x^n$. Is my understanding correct that
$$ R/x \cong \mathbb{R} $$
and accordingly, as scalars divide all polynomials,
$$R/(R/x) \cong \{1\}$$
...
0
votes
0answers
34 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?
1
vote
2answers
47 views
For polynomials $f, g$ and $\gcd(f(X),g(X))=d(X)$ then $\gcd(f(X+a),g(X+a))=d(X+a)$
Suppose for polynomials f, g in $\mathbb{Q}[X]$ it holds that
$$\gcd(f(X),g(X))=d(X)$$
What we also want to now prove is that for $a \in \mathbb{Q}$: $$\gcd(f(X+a),g(X+a))=d(X+a)$$
So the ...
2
votes
1answer
44 views
How many polynomials of degree $n$ in $\mathbb{Z}/3\mathbb{Z}$
We are given the polynomial ring $\mathbb{Z}/3\mathbb{Z}$. We are asked to determine how many polynomials of degree $n$ there are. First of all, the possible coefficients are $0, 1,2$, if we have a ...
2
votes
0answers
34 views
Prove that any polynomial with integer coefficients must have a composite number in its image. [duplicate]
Let $f \in \mathbb{Z}[x]$ be a polynomial of degree at least $1$. Prove that there is $n \in \mathbb{Z} $ such that the corresponding polynomial function $f(n)$ is not a prime.
I think that the ...
4
votes
2answers
74 views
Show that $\mathbb{Z}_5[x]/(x^2+x+1)\cong\mathbb{Z}_5[x]/(x^2+x+2)$
I think this problem is from Gallian, prof couldn't solve it. Notice that both polynomials have no roots. I tried to construct an onto homomorphism $\varphi:\mathbb{Z}_5[x]\to\mathbb{Z}_5/(x^2+x+2)$ ...
3
votes
0answers
40 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 ...
2
votes
2answers
114 views
Given any commutative ring $R$ with unity, $R[X]$ has infinitely many maximal ideals.
Hope this isn't a duplicate.
I was trying to answer the following questions:
(i) Let $k$ be any field. Then prove that $k[X]$ has infinitely many maximal ideals.
(ii) Using (i) prove that, given ...
6
votes
2answers
64 views
$\Bbb R[X,Y]/(F) \cong \Bbb R[Z]$ and $F_X G_Y - G_X F_Y \in \Bbb R^*$
Hope this isn't a duplicate.
I was trying to solve the following problem :
Let $F,G \in \Bbb R[X,Y]$ satisfy $\Bbb R[F,G]= \Bbb R[X,Y]$. Prove that :
(i) $\Bbb R[X,Y]/(F) \cong \Bbb R[Z]$ , for ...
6
votes
1answer
94 views
Factoring $x^n + 1$.
By the Fundamental Theorem of Algebra, every polynomial of degree $n$ can be factored into a product of $n$ linear polynomials.
As an example, since the polynomial $ x^5 +1$ has the five complex ...
0
votes
1answer
28 views
Polynomial Evaluation. How to (formally) substitute X with a Matrix
for simplicity a specific case:
$f:=(a_i)_{i \in \mathbb{N}} \in \mathbb{R}[X]$
$A \in \mathbb{R}^{4\times 4}$
In one of our assignments f is evaluated as $\sum_{i=0}^{\deg(f)} a_iA^i$.
How is ...
1
vote
1answer
63 views
Show that $\mathbb{Z}_3[i]$ is a field
I know that $\dfrac{\mathbb{Z}_3[x]}{\langle x^2+1\rangle}$ is isomorphic to $\mathbb{Z}_3[i]$, does this help me prove that $\mathbb{Z}_3[i]$ is a field?
$\langle x^2+1\rangle$ is the ideal ...
2
votes
1answer
75 views
Show that every nonzero prime ideal in $\mathbb{C}[x, y]/(x^2 - y^2 - 1)$ is maximal [duplicate]
Show that the ring $A := \mathbb{C}[x, y]/(x^2 - y^2 - 1)$ is an integral domain. Further show that every nonzero prime ideal in A is maximal.
I proved that $A$ is an integral domain by showing that $...
2
votes
1answer
44 views
How do the proper ideals of $\Bbb Q[x]/ \langle x^3 \rangle$ look like?
How do the proper ideals of $\Bbb Q[x]/ \langle x^3 \rangle$ look like?
I know that the elements of it look like $p(x)$ modulo $x^3$. So the ideals of it look like $\langle ax^2+bx+c\rangle $ where $...
4
votes
1answer
45 views
Why isn't this criterion for determining irreducibilty working?
I have learned this criterion for irreducibility of polynomials:
Let $R$ be an integral domain, let $I$ be a proper ideal of $R$, and let $p(x)$ be a non-constant monic polynomial in $R[x]$. If the ...
0
votes
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 ...
1
vote
2answers
46 views
Trouble with a step in proof of "$R$ is a U.F.D. $\implies R[x]$ is a U.F.D.
I am reading a proof of this in Dummit & Foote (Chapter $9.3$, Thm. $7$).
The proof uses Gauss' Lemma:
Gauss' Lemma: Let $R$ be a U.F.D. and let $F$ be its field of fractions. If $p(x) \in {R}...
7
votes
3answers
317 views
Show that the ideal generated by two polynomials is actually equivalent to the ideal generated by a single polynomial
I've been asked to show that the ideal $I = \langle x^3 - 1, x^2 - 4x + 3\rangle$ in $\mathbb{C}[x]$ is equivalent to $I = \langle p(x)\rangle$ for some polynomial $p(x)$.
Now, I'm not quite sure ...
