0
votes
0answers
9 views

Are these computational models equivalent?

Let $f : X \to Y$ be a problem that you want to compute. Say we have an $O(1)$-computable maps, $\phi, \psi$, such that $X \xrightarrow{\phi} (\Bbb{Z}_n)^k \xrightarrow{\psi} Y$. After all, ...
1
vote
0answers
13 views

Creating Polynomial

By relative prime factor theorem $$R = (Zm,+,.)$$ where R is the ring structure the input is $e_0 = 0$ and $e_1=1$ output is $$S_0 = { k : \gcd(m,k)>1 }$$ $$S_1 = { k : \gcd(m,k) = 1}$$ Now ...
1
vote
0answers
72 views

Calculate the primary decomposition

Consider the polynomial ring $R=K[x_1,\ldots, x_8]$ over field $K$. Set $\mathfrak{p}_1=(x_1, x_2, x_5, x_6)$, $\mathfrak{p}_2=(x_3, x_4, x_7, x_8)$ and $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
5
votes
3answers
90 views

Prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$

I want to prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$. Eisenstein's criterion doesn't apply here, so I think the simplest method would be to use the Rational Roots Test, right? If I can use ...
5
votes
1answer
54 views

Topological closure of ideal in $A[[T]]$ - Proposition 1.3.7 in Liu

In Proposition 1.3.7 of Liu's book, one proves that if a ring $A$ is noetherian then so is $A[[T]]$. We take an ideal $I$ of $A[[T]]$ and prove that there exist $F_1,\ldots,F_m\in I$ such that for all ...
2
votes
2answers
47 views

Reducibility of a Cyclotomic Polynomial under the ring homomorphism $\mathbb{Z} \rightarrow \mathbb{F}_p$

I'm working through the following question: Question Reference: Oxford Part I Paper B2 2003 Find the monic polynomial $f \in \mathbb{Z}[X]$ whose roots are the complex primitive ...
4
votes
4answers
97 views

Want to prove that some $\mathbb R[x]$-Module has no basis

So here is my question, Consider the $\mathbb R[X]$-module $\mathbb R[X,X^{-1}]$ i.e the $\mathbb R[x]$-module of all Laurent-Polynomials. I want to show that is module is not free i.e it has no ...
0
votes
1answer
26 views

Proving $(φ(x)\cdot ψ(x)) \cdot ω(x)=φ(x) \cdot (ψ(x)\cdot ω(x))$ where $φ,ψ,ω$ are polynomials on a ring $R[X]$

If I take $3$ random polynomials $φ,ψ,ω$ on a ring $R[x]$, I'm trying to prove associativity which is very obvious. But I have trouble on the algebra part with the sums. I know that given $2$ ...
1
vote
2answers
61 views

algebraic integer $\alpha$ + polynomial relation $\beta$ and $\alpha$ $\Rightarrow$ $\beta$ algebraic integer.

Assume $\beta$ can be expressed in terms of polynomial relation in $\mathbb{Z}[\alpha]$. Where $\alpha$ is an algebraic integer (i.e. $\alpha$ is the root of a polynomial in $\mathbb{Z}[X]$. How can ...
4
votes
0answers
49 views

The ring of homogeneous polynomials

I think I found an error in my textbook, but I am not completely sure. The book is Hulek, Elementary algebraic geometry, pag. 73. There is a theorem showing that $U_i$ and the affine space ...
1
vote
1answer
33 views

Polynomials over a field with characteristic $0$ is square free implies it's coprime with its formal derivative

Let $F$ be a field with characteristic $0$, $f \in F[t]$ the polynomial ring over $F$. Show that $f$ is square free implies $ f, f'$ are relatively prime. I know this is actually an if and only if ...
0
votes
1answer
16 views

Degree 3 polynomial with coef in a field K: question on roots in a algebraic closure

I am asked the following question: Consider a field $K$ with characteristic different from 2 and 3, and the polynomial $f(t) = t^3 + pt + q \in K(t)$ with three distinct roots $\alpha_1, \alpha_2, ...
3
votes
1answer
100 views

Number of common zeros of two quadratic polynomials in ${\Bbb C}[t,x]$

The following theorem is in Artin's Algebra(2nd edition): Theorem 11.9.10 Two nonzero polynomials $f(t,x)$ and $g(t,x)$ in two variables have only finitely many common zeros in ${\Bbb C}^2$, ...
0
votes
2answers
44 views

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. [duplicate]

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. Not sure how to proceed with this problem. I usually use Chegg, but Chegg doesn't have the solution for this problem. ...
5
votes
1answer
97 views

Exercise from Atiyah-Macdonald, Chapter 1, 2.iv)

Let $A$ be a ring and let $A[x]$ be the ring of polynomials in an indeterminate $x,$ with coefficients in $A.$ Let $f=a_0 + a_1x+\cdots+a_nx^n \in A[x].$ $f$ is said to be primitive if ...
2
votes
0answers
38 views

Properties of cyclotomic polynomial

Assume first that $p$ a prime divides $n$. I have to show that $\Phi_{np}(X)=\Phi_n(X^p)$. Here is what I tried: Suppose $\eta_i$ are roots of $\Phi_{np}(X)$ so $\eta_i=\text{exp}(\frac{2\pi i ...
3
votes
1answer
72 views

Ring of Polynomials Commutative?

Can $k[x_1,...,x_n]$, the ring of polynomials with coefficients $\in k$ where $k$ is a field, ever be a non-commutative ring?
3
votes
2answers
89 views

What is the proof of the single factor theorem over an arbitrary commutative ring?

Theorem (Single factor theorem) Let $R$ be a commutative ring, and let $P\in R[X]$, where $R[X]$ is the polynomial ring over the indeterminate $X$. Suppose $P(\alpha)=0$. Then $(X-\alpha)$ divides ...
1
vote
1answer
75 views

Can this quick way of showing that $K[X,Y]/(Y-X^2)\cong K[X]$ be turned into a valid argument?

I've been trying to show that $$ K[X,Y]/(Y-X^2)\cong K[X] $$ where $K$ is a field, $K[X]$ and $K[X,Y]$ are the obvious polynomial rings over the indeterminates $X$ and $Y$ and $(Y-X^2)$ is the ...
0
votes
1answer
36 views

Given a polynomial $p(x)$ in $\mathbb Z_6[x]$, it is possible to construct a ring $R$ such that $p(x)$ has a root in $R$.

Prove or disprove: Given a polynomial $p(x)$ in $\mathbb Z_6[x]$, it is possible to construct a ring $R$ such that $p(x)$ has a root in $R$. For this exercise I think about complex numbers. ...
0
votes
1answer
76 views

Kernel and direct sum

Let $R=k[x_1,\ldots,x_7]$ be a polynomial ring over field $k$ and $I=\bigcap_{i=1}^4 \mathfrak{p}_i$ where $\mathfrak{p}_1=(x_1,x_3,x_5,x_6), \mathfrak{p}_2=(x_1,x_3,x_4,x_6), ...
1
vote
1answer
66 views

About the notation $\mathbb{Z}[x]/(f(x),p)$

Let $f(x)\in \mathbb{Z}[x]$ be a polynomial and $p$ be a prime. What does the notation $\mathbb{Z}[x]/(f(x),p)$ mean? Is it $\mathbb{Z}/p\mathbb{Z}[x]/(f(x))$ ?
1
vote
1answer
77 views

Identifying some quotient rings

How come that $k[w,z]/(w^2+z,w^3 z^2)\cong k[w]/(w^7)$? Also why is $(xz,w)=(x,w)\cap(z,w)$ in the polynomial ring in 3 variables? what are the rules of ideal calculus making these results evident?
0
votes
1answer
145 views

Subrings of polynomial rings over the complex plane

I have the following questions: (i) must every subring of the polynomial ring in two variables over the complex plane, containing the complex plane itself, be Noetherian? (ii) Are there Noetherian ...
0
votes
0answers
31 views

polynomial ring over a domain is also a domain + every non-constant polynomial over a certain field has root?

Show that if $R$ is a domain, so is the polynomial ring $R[t]$. In particular, show that there exists fields properly containing the complex numbers. Does this field field has the property that every ...
2
votes
3answers
63 views

A question about degree of a polynomial

Let $R$ be a commutative ring with identity $1 \in R$, let $R[x]$ be the ring of polynomials with coefficients in $R$, and let the polynomial $f(x)$ be invertible in $R[x]$. If $R$ is an integral ...
2
votes
2answers
67 views

Polynomial in two variables with zero constant coefficient form principal ideal?

Let $F$ be a field, and $F[x,y]$ the ring of polynomials in $x,y$. Let $J$ be the subset of all polynomials $P(x,y)$ in $F[x,y]$ such that $P(0,0)=0$. Then $J$ is an ideal. Is $J$ a principal ideal?
0
votes
3answers
28 views

Divisibility of polynomials in $\mathbb{Z}_n[x]$

For what values of $n$ is $x^2+1$ a factor of $x^5+5x+6$ in $\mathbb{Z}_n[x]$? I know how to divide in $\mathbb{Z}[x]$ (with long division), but what should I do here with $\mathbb{Z}_n[x]$, and it's ...
0
votes
2answers
98 views

How to show that $x$ becomes a root of $p(x)$ in $F[x]/(p(x))$

$F$ is a field, $p(x)$ is irreducible polynomial at $F[X]$. $K=F[X]/\left<p(x)\right>$. For every $a\in F$ we will mark: $\bar{a}=\left<p(x)\right>+a$. Now, the question is: How do I show ...
0
votes
2answers
88 views

How to find a polynomial product that give me $x^6+1$

I need to find a polynomials product that give me $x^6+1$ at $\mathbb{R}[X]$ and at $\mathbb{C}[X]$. I need that the product will be of irreducible polynumials... Thank you!
0
votes
1answer
41 views

coprime elements

Let $R$ be a ring, then two elements $I,J$ are coprime, if $RJ+RI=R$ or in other words, if there exist $r_1,r_2 \in R$ such that $r_1I+r_2J=u$, where $u$ is a unitity in $R$. Now let $\mathbb{Q}$ be ...
-1
votes
4answers
77 views

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$. How to prove? I really have no idea... Thank you a lot.
0
votes
1answer
60 views

$gcd(a,b)$ in a UFD subring is not a greatest common divisor in the ring

Give a counterexample that $R$ is a unique factorization domain but not a principal ideal domain, $S$ is a ring containing $R$, such that $a,b\in R$, $gcd(a,b)$ in $R$ is not a greatest common ...
1
vote
1answer
36 views

irreducible elements of polynomial rings

Let $p$ be a prime integer. For $x\in\mathbb{Z}$, let $x'$ be the remainder of $x$ when divided by $p$. Let $\sum_{i=0}^{n}a_iX^i\in \mathbb{Z}[X]$ with $p$ does not divide $a_n$ in $\mathbb{Z}$. Then ...
1
vote
2answers
175 views

Should the sum of zero divisors also a zero divisor?

In a general ring $A$ (commutative with $1$), should the sum of two zero divisors also a zero divisor? Could anyone give a proof or a countexample? Moreover, consider the polynomial ring $A[x]$, ...
2
votes
0answers
50 views

Different elements in a factor ring

Studying for my algebra exam I found the following problem, which I'm not sure how to solve Let $f = X^2 + 1 \in \mathbb{F}_5[X]$, $R = \mathbb{F}_5[X]/\langle f \rangle$ and $\alpha = X + \langle ...
0
votes
0answers
95 views

Let $f:R\longrightarrow S$ be a surjective ring homomorphism. If $R$ is PID, then $S$ is PID.

Let $f:R\longrightarrow S$ be a surjective ring homomorphism. If $R$ is PID, then $S$ is PID. I think I have proved this: Let $J$ be an ideal of $S$. Then $f^{-1}(J)=(a)$ is a principal ideal of ...
0
votes
1answer
55 views

Let $R$ be an integral domain and $I$ be a prime ideal of $R$. If $R/I$ is a Euclidean domain, will $R$ be a unique factorization domain?

Let $R$ be an integral domain and $I$ be a prime ideal of $R$. If $R/I$ is a Euclidean domain, will $R$ be a unique factorization domain? I have no idea to prove or disprove this... should I prove ...
0
votes
2answers
44 views

Quotient group element is a unit

I'm studying up for my algebra exam, and I'm not exactly sure how to solve a problem like the following Let $f = X^2 + 1 \in \mathbb{F}_5[X]$, $R = \mathbb{F}_5[X]/\langle f \rangle$ and $\alpha = ...
4
votes
3answers
80 views

Show that $\alpha^2 + \alpha - 1$ is a zero divisor in $R$

Studying for my algebra exam and looking through old exam exercises I came across the following problem Let $f = X^4 + 1$, $g = X^2 + X - 1 \in \mathbb{F}_3[X]$ and $\alpha = X + \langle f \rangle ...
1
vote
2answers
42 views

Factor ring of polynomial

$F[x]$ is a polynomial ring over a certain field $F$. $J$ is an ideal of $F$, $J = (f(x))$. I need to prove that if the polynomial $f(x)$ has a multiple root the factor ring $F[x]/J$ is not a field. ...
1
vote
1answer
29 views

Showing $f(x^{p_1}) \mid f(x^{p_1 p_2})$ Given that $f(x) \mid f(x^{p_1}), f(x^{p_2})$

Hypothesis: Let $f = a_0 + a_1x + \ldots + a_nx^n \in \mathbb{Z}[x]$. Suppose $f(x) \mid f(x^{p_1})$ and $f \mid f(x^{p_2})$ for $p_1$ and $p_2$ two positive prime integers. Goal: Show that ...
6
votes
3answers
99 views

Minimal polynomial of $\alpha^2$ given the minimal polynomial of $\alpha$

Given that $\alpha$ is a root (in the field extension) of the irreducible polynomial $X^4+X^3-X+2\in\mathbb{Q}[X]$, I have to find the minimal polynomial of $\alpha^2$. I am thinking about this for a ...
2
votes
2answers
35 views

Find $c_1,c_2,c_3\in\mathbb{Q}$ such that $(1+\alpha^4)^{-1}=c_1+c_2\cdot\alpha+c_3\cdot\alpha^2$ in $\Bbb Q(\alpha)$.

Let $\alpha\in \bar{\mathbb{Q}}$ a root of $X^3+X+1\in\mathbb{Q}[X]$. So this is the minimal polynomial of $\alpha$ because its irreducible in $\mathbb{Q}[X]$. I had to find the minimal polynomials of ...
4
votes
2answers
102 views

Nilpotent/invertible polynomial over commutative ring. [duplicate]

Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial over a commutative ring $R$. Prove that (a) $p$ is unit in $R[x]$ iff $a_0$ is unit and $a_1,a_2,\ldots,a_n$ are nilpotent in ...
1
vote
1answer
65 views

What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$

I'm doing some exercises to prepare for my exam: What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$. I've no idea how to tackle this ...
-2
votes
2answers
46 views

Frobenius maps exist/do not exist for integers?

Does there exist an infinite ring $R$ such that $(x+y)^b=x^b+y^b$, and similarly for $2$ other odd primes $a,c$; in which $\Bbb{Z}$ can be embedded as a ring? I have no idea where to begin. Maybe ...
-1
votes
1answer
35 views

Does there exist a infinite ring in which there are $3$ Frobenius homs?

Does there exist an infinite ring $R$ such that $(x + y)^b = x^b + y^b$, and similarly for $2$ other odd primes $a,c$? Or what's the best that can be done?
2
votes
1answer
36 views

What more can we say about $\mathbb{Z}[x]/(x+1)$?

Related to this question here Adjoining elements to $\mathbb{Z}$ given a set of generators: I want to determine the structure of $R'$ obtained by adjoining $\alpha$ to $\mathbb{Z}$ with generators ...
7
votes
1answer
74 views

Short method to prove irreduicibility of $x^7+21x^5+35x^2+34x-8$ over $\Bbb Q$?

I am given a task to prove that polynomial $f=x^7+21x^5+35x^2+34x-8$ is irreducible over $\Bbb Q$? In my algebra course we learnt reduction and Eisenstein criterion. Eisenstein doesn't seem to work ...