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".
1
vote
0answers
27 views
If $\tilde{g}(\tilde{g}-1) \in \sqrt{I}$ then there exists $g \in I$ such that $g(g-1) \in I$
I am trying to prove the following. Let $I$ be an ideal of the polynomial ring $R=k[T_1, \ldots, T_n]$ and $\sqrt{I}$ it's radical ideal. If I have an element $\tilde{g} \in R$ such that $\tilde{g}(\...
3
votes
0answers
26 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
24 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,
$$R/(R/x) \cong \{1\}$$
...
-1
votes
0answers
6 views
How to show $\beta$'s minimal polynomial $p(x)\in K[x]$ takes its coefficients in $A$? [duplicate]
Let $A$ be an integral domain which is integrally closed, $K$ its field of fractions, $L|K$ a finite field extension. If $\beta\in L$ is integral over $A$, how to show its minimal polynomial $p(x)\in ...
0
votes
0answers
24 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
31 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
30 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
30 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
67 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
33 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
68 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
59 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
88 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
25 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
59 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
60 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
42 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
44 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
42 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}...
5
votes
3answers
221 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 ...
2
votes
2answers
50 views
If $R$ is a commutative ring with identity then $R[x_1, x_2, …, x_n]$ is not a P.I.D.?
I am working on problem $7$, section $9.1$ in Dummit & Foote:
Let $R$ be a commutative ring with $1$. Prove that a polynomial ring in more than one variable over $R$ is not a principal ideal ...
0
votes
0answers
31 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 $F[x, y]$.
$a)$ ...
5
votes
3answers
161 views
How do we know that automorphisms on polynomials have a polynomial like form?
Let $F$ be a field and $\sigma:F[x]\to F[x]$ be automorphism, $\sigma(a) = a$ for all $a\in F$. I'm supposed to show that $\sigma(f(x)) = f(ax+b)$ for some $a\not = 0$ and $b$ in $F$.
Now I've got a ...
4
votes
0answers
27 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 $$f_i:=\begin{cases}1\text{ if }i=1,\...
0
votes
2answers
32 views
Finite field adjunction - isomorphism that maps from the field into a polynomial ring factored over an ideal
A field $(K, +, 0, *, 1)$ is given, and it is a finite field with $25$ elements. $P$ is the prime field over K. Further it was given that a polynomial $r(x):=x^2+x+1 \in P[x]$ is irreducible.
Now, ...
0
votes
1answer
28 views
Surjective ring homomorphism - cardinality of the domain
I've been doing some exam preparation examples, and I wanted to ask if my idea to this example makes sense - at least if I am on the right path.
A field $P \cong \mathbb{Z}_7$ and a polynomial ring $...
1
vote
2answers
54 views
Determine all the ideals of $\mathbb{Q}[x,y]$ that contain the ideal $\langle x^2,y^2,xy\rangle$.
Determine all the ideals of $\mathbb{Q}[x,y]$ that contain the ideal $I=\langle x^2,y^2,xy\rangle$.
Are these all:
$$\langle x,y\rangle$$
$$I$$
$$\mathbb{Q}[x,y]$$
?
EDIT....
$$\langle x^2,y \...
2
votes
1answer
43 views
Proving that $\mathbb{Z}[x]/(x^3-2x+1)$ is not an integral domain and that x is a unit
Proving that $E=\mathbb{Z}[x]/(x^3-2x+1)$ is not an integral domain and that $x$ is a unit.
Let $r=(x^3-2x+1)$
So to show that $\mathbb{Z}[x]/(x^3-2x+1)$ is not an integral domain I must find a $p(x)...
2
votes
1answer
36 views
Let f (x) be a nonconstant element of Z[x]. Prove that f (x) takes on infinitely many values in Z.
This is a homework problem so no need to give me an immediate answer.
My general plan has been to try to prove that f(x) cannot have both an upper bound and a lower bound.
One strategy I tried was ...
1
vote
1answer
23 views
Ideal generated by $x_1 - a_1, \dots, x_k - a_k$ is prime in polynomial ring over integral domain
I would like to show the ideal $P_k = \langle x_1 - a_1, \dots, x_k - a_k \rangle$ of $R = F[x_1,\dots,x_n]$, where $F$ is a field, $k \leq n$ and $a_i \in F$, is a prime ideal.
In the case where all ...
0
votes
0answers
29 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
votes
0answers
23 views
numerical calculation of principal ideal generator in polynomial ring
I will first state my problem and explain where I am at with the problem.
Let $I$ be a principle ideal generated by a single, multivariate polynomial $p \in \mathbb{C}[z_1,z_2,z_3]$.
I estimate a ...
1
vote
1answer
148 views
Finding all the ideals of $\mathbb{Z} [x]/(2, (x^3+1))$?
This is from Dummit and Foote (Section 9.2):
7. Determine all the ideals of the ring $\mathbb{Z} [x]/(2, (x^3+1))$.
This is my attempt to understand what's going on:
My plan is to find a nice ...
1
vote
1answer
161 views
Find all irreducible polynomials of degrees $2$ and $3$ in $\Bbb{Z}_{2}[x]$.
Find all irreducible polynomials of degrees $2$ and $3$ in $\Bbb{Z}_{2}[x]$.
I tried to follow (https://math.stackexchange.com/q/32416)'s answer but this part in their answer I do not understand:
...
2
votes
1answer
44 views
Noether polynomial ring over $2\Bbb Z$
Given a definition of $R=2\Bbb Z$ prove that ring $R[x]$ is not noether.
I assume that the proof should be based on the fact that a ring is noether if and only if any ideal is finitely generated. ...
1
vote
0answers
53 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 ...
4
votes
2answers
99 views
Counting irreducible polynomials over finite fields [duplicate]
How many irreducible polynomials in $Z_2[x]$ of degree $3$?
I have discussed this with my friend before and we found that $x^3 + x^2 + 1$ and $x^3+x+1$ are the two said polynomials which irreducible....
0
votes
0answers
30 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 $\mathbb{Z}_n$) ...
3
votes
1answer
27 views
notation in congruence relation
hi there i was looking through my lecture notes and i'm struggling to understand a particular piece of notation the vertical line | and i was wondering if you could explain its meaning
$$f \sim g \...
2
votes
1answer
85 views
Descending Chain Condition
How to show that $k[x]$, polynomial ring over $k$ where $k$ is a field, does not satisfy the Descending Chain Condition.
The Descending Chain Condition is if $A_1 \supseteq A_2\supseteq A_3....... $ ...
1
vote
1answer
128 views
Polynomial ring over finite field - inverting a polynomial non-prime
I'm trying to recreate the wiki's example procedure, available here:
https://en.wikipedia.org/wiki/NTRUEncrypt
I've run into an issue while attempting to invert the polynomials.
The SAGE code below ...
0
votes
2answers
42 views
Prove that $\mathbb{Z}_{5}[x]/(x^2+1)$ is isomorphic to $\mathbb{Z}_{5} \times \mathbb{Z}_{5}$.
I remember trying this problem a while ago and was unable to prove it. I think my idea was to create a surjective homomorphism from $\mathbb{Z}_{5}[x]$ to $\mathbb{Z}_{5} \times \mathbb{Z}_{5}$ and ...
1
vote
2answers
24 views
What are the coefficients of $x^2+2\in(\mathbb{Z}/\mathbb{Z}4)[x]?$
If $$(\mathbb{Z}/\mathbb{Z}4) = \{\bar{0}, \bar{1}, \bar{2}, \bar{3}\}$$
How can it be possible that $x^2+2 \in (\mathbb{Z}/\mathbb{Z}4)[x]$?
In other words, how can the 2 value in $x^2+2$ be ...
0
votes
1answer
67 views
The 1-affine space is not isomorphic to the 1-affine space minus one point
I have to prove that $\Bbb{A}^1$ is not isomorphic to $\Bbb{A}^1-\{0\}$ . Apparently one does this by showing that the corresponding coordinate rings are not isomorphic, but I have $I(\Bbb{A}^1-\{0\})=...
0
votes
0answers
37 views
Proof that these polynomials form a Gröbner basis
Given $n$ points $p_0, \ldots p_{n-1}$ in the plane $\Bbb Q^2$ and the ideal $I_n$ of all polynomials ($\in R = Ring(\Bbb Q,[x,y])$, the ring of polynomials with rational coefficients in the variables ...
2
votes
1answer
169 views
Prove that the field $k(x)$ of rational functions over $k$ in the variable $x$ is not a finitely generated $k$-algebra.
I am working through Chapter 15 of Dummit and Foote's Abstract Algebra text, and I am stumped on how to prove the following (Exercise 3):
Prove that the field $k(x)$ of rational functions over $k$ in ...
1
vote
2answers
68 views
For what $k$ is $g_k\circ f_k$ invertible?
Let $\ f_k:\Bbb R[x]_{\le1}\to\Bbb R^3, g_k:\Bbb R^3\to\Bbb R[x]_{\le1}$ be linear maps such that $f_k(x-1)=\left( {\begin{array}{*{20}{c}}
{{0}} \\
k \\
{1}
\end{array}} \right),\ f_k(2-x)=...
3
votes
1answer
687 views
Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain. [duplicate]
I want to prove that for a ring $R$, $R[x]$ is an integral domain if and only if $R$ is an integral domain.
I have one direction of the proof ($R$ an integral domain implies $R[x]$) an integral ...
2
votes
1answer
74 views
Identifying the ring $R=\mathbb{Z}_{9}[x]/(x^2-3,3x)$
I want to know if there is a simple form of the ring
$$R=\mathbb{Z}_{9}[x]/(x^2-3,3x)$$
I tried to start with the equations $3x\equiv0$ and $x^2\equiv 3$. So, $3x^2\equiv 0$ and $3x^2\equiv 9$. ...
