6
votes
2answers
84 views

Minimal polynomial: is $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1$?

I was wondering about the minimal polynomial of real number $$u=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$$ over field $\mathbb{Q}$. As you can see here, I worked out that $u$ is a root of monic ...
1
vote
0answers
64 views

Necessary and sufficient conditions for $\rm P \neq NP$ maybe?

Please review the $\rm P \neq NP$ problem here. I'm working on an algebraic approach to this problem, and all my notes are currently here. Conjecture 1 For all $f \in F[x_1, \dots, x_k]$, a minimal ...
2
votes
0answers
39 views

Irreducibility of a Polynomial over Q

How do I show that for any odd prime $p$ the polynomial $f(x)=x^p-9$ is irreducible over $\mathbb Q$?
1
vote
1answer
33 views

Comparing coefficients in finite field

We start with the wrong proof of the following theorem: $p| \binom{p}{k}$ for a prime $p$ and $0<k<p.$ Proof: $(1+x)^p \equiv 1+x \equiv 1+x^p \pmod{p}$ by Fermat's little theorem. Comparing ...
1
vote
0answers
77 views

Too many independent cubic polynomials in an ideal $I\subset \mathbb C[x,y,z]$

Let us consider the ideal $I=(x^2-x,y,xz)\subset \mathbb C[x,y,z]$. I want to prove that $I$ contains (exactly) $5$ linearly independent polynomials of degree $3$. In three variables, we have ...
1
vote
0answers
25 views

Find all integers $m$ and positive integers $n > 1$ so that $m + \sum_{k=1}^n x^k/k!$ has a rational root

If $m = 1$, then $m + \sum_{k=1}^n x^k/k!$ has no rational root for $n > 1$. And clearly the polynomial has a rational foot for all integers $m$ if $n = 1$. So, besides those cases, for what ...
2
votes
1answer
38 views

Show that $(x-a,x-b)=1$

Knowing that $K$ is a field, $a,b \in K$ different from each other,show that $x-a,x-b$ co-primes. We suppose that $\exists f(x) \in K(x)$ such that: $f(x)|x-a$ and $f(x)|x-b$ Then $\deg f(x) \leq ...
1
vote
2answers
51 views

Is the composition of irreducible polynomials again irreducible

I've been pondering this since yesterday. Is it true that given two irreducible polynomials $f(x)$ and $ g(x)$ will $f(g(x))$ or $g(f(x))$ be irreducible?
2
votes
1answer
39 views

Show there exists another polynomial with specified roots.

Let $\alpha$ be a complex number. Suppose there exists a a monic polynomial $f(x) \in \mathbb{Z}[x]$ such that $f(\alpha)=0$. Show that there exists a monic polynomial $g(x) \in \mathbb{Z}[x]$ such ...
1
vote
1answer
51 views

Is it possible that the zeroes of a polynomial form an infinite field?

Let $K/F$ be a finite field extension and suppose that $F$ is infinite. Is it possible to have a nonzero polynomial $p \in K[x_1,...,x_n]$ that vanishes in $F^n$?
0
votes
1answer
25 views

Greatest common divisor of ring elements

Consider the ring $\mathbb Q[x]$. (a) Suppose that $a(x) = (x+1)^3(x-1)^4(x+2)$ and $b(x) = (x+1)^2(x+2)^3(x-3)^4$. What is the $\gcd (a(x),b(x))$? (b) Suppose that $c(x) = (x^2-1)^4(x^2+3x+2)$. ...
10
votes
1answer
50 views

Construction of a polynomial

Let $ m \in \mathbb{N}$ be fixed and $q=p^n$ (a variable prime power) for $n \in \mathbb{N}$ and $p$ prime. We define $$c_m=|\left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, monic, ...
6
votes
1answer
31 views

Number of monic polynomials = $q^n$?

In the situation $q=p^k$ with $p$ prime and $k \in \mathbb{N}$ I have the following question: Why is the number of monic polynomials of degree $n$ in $\mathbb{F}_q[X]$ $$q^n \ ?$$
10
votes
1answer
143 views

Calculating a strange algebraic limes

I have a problem with calculating a strange limes: Let $ m \in \mathbb{N}$ be fixed and $q=p^n$ (a variabel prime power) for $n \in \mathbb{N}$ and $p$ prime. We define $$c_m=|\left\lbrace f \in ...
5
votes
1answer
78 views

Difficult algebraic problem - irreducible polynomials

Let $p$ be a prime number and $q=p^n$ with $n \in \mathbb{N}$. We define the polynomial$$ F_m:=\prod \left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, nomic}, \text{deg}(f)=m \right\rbrace $$ ...
10
votes
1answer
117 views

Cyclotomic polynomial - coefficient

For a polynomial $f=X^n+a_1X^{n-1}+\ldots+a_n \in \mathbb{Q}[X]$ we define $\varphi(f):=a_1 \in \mathbb{Q}$. Now I want to show that for the $n$th cyclotomic polynomial $\Phi_n$ it holds that ...
0
votes
1answer
49 views

Order of element of multiplicative group of finite field mod polynomial

If $K$ is a finite field of size $q$ and $f$ is a degree $n$ polynomial in $K[x]$, then we can form the quotient field by modding out this polynomial. Elements of this quotient field are of the form ...
7
votes
2answers
443 views

How does Hilbert's Nullstellensatz generalize the “fundamental theorem of algebra”?

What is Hilbert's Nullstellensatz in the sense of the generalization of "fundamental theorem of algebra"? I've seen that in some texts it was referred to as the generalization of the fundamental ...
2
votes
1answer
51 views

A question about polynomials in $K[x_1,x_2,…,x_n]$ and there permutations

Let $K$ be a field. Let $n$ be a positive integer and $P$ be a non-symmetric polynomial in $K[x_1,x_2,...,x_n]$. $S_n$ acts on $K[x_1,...,x_n]$ in an obvious way. Let $P_1,P_2,...,P_r $ be the ...
1
vote
1answer
24 views

GCD for multivariable polynomial ring

I'm reading Lectures on Modules and Rings by T. Y. Lam. It's on page 32 of the book, example 2.19A. It reads: (2.19A) Example. Let $k$ be a field. Then in the commutative polynomial ring $R = ...
0
votes
1answer
47 views

Homogeneous polynomial in a homogeneous ideal

Let $f$ be a non-zero homogeneous polynomial in a homogeneous ideal generated by homogeneous elements $g_1,\ldots, g_s$. Suppose $f= h_1g_1 +\cdots+h_sg_s$. Is it necessary that $\deg(f)=\deg(h_ig_i)$ ...
3
votes
1answer
65 views

a polynomial about continuous function

Let $\{a_i(x):\mathbb{R}\rightarrow \mathbb{C}\}$ be continuous functions, does there exist some continuous functions $\{\lambda_i(x)\}$ such that $$a_{n-1}(x) y^n+a_{n-2}(x) y^{n-1}+\cdots ...
2
votes
1answer
60 views

Relation between divisibility of polynomials in different rings, $h | f$ in $\mathbb{Z}[x], \mathbb{Z}/p^k\mathbb{Z}[x]$ and $\mathbb{F}_p[x]$

Let $p$ be a prime, $k$ a positive integer. Let $f,h \in \mathbb{Z}[x]$ be polynomials such that $h | f \mod p^k$ in $ (\mathbb{Z}/p^k\mathbb{Z})[x]$ $h \mod p$ is irreducible in $\mathbb{F}_p$ ...
2
votes
2answers
114 views

Help in this proof in Lang's Algebra book

I'm trying to understand this part of the proof: I didn't understand why not all coefficients of $f_2,\ldots,f_n$ can lie in the maximal ideal, maybe I'm forgetting something, it should be a very ...
1
vote
0answers
29 views

Express symmetric polynomial $\prod_{i < j} (X_i+X_j)$ in terms of elementary symmetric functions

Exercise: Define a polynomial $\Sigma(X_1,\ldots,X_n)$ as \begin{align*} \Sigma(X_1,\ldots,X_n) = \prod_{i < j} (X_i+X_j) \end{align*} This is a symmetric polynomial, quite clearly. I want to ...
3
votes
2answers
52 views

$f$ has root $\alpha$, then $f = (X-\alpha)g$ for some $g$

I need some help with the following problem: Suppose $R$ is a unique factorisation domain and $f \in R[X]$ such that $\deg f > 0$ and $f$ has a root $\alpha \in R$. Then $f = (X-\alpha)g$ for some ...
0
votes
0answers
21 views

Can the cubic mean- geometric mean inequality by a sum of 8 weighted squares?

Does there exist 8 trivariate polynomials $f_1, \ldots, f_8$ and a function $s:\{1, \ldots, 8\}\to \{1, x,y,z\}$ such that $x^3 + y^3 + z^3 - 3xyz = s(1)f_1(x,y,z)^2 +\cdots + s(8)f_8(x,y,z)^2$ for ...
4
votes
2answers
44 views

Does there exist bivariate polynomials $p$ and $q$ such that $p(x,y)^2 = q(x, y)^2 ( x^2 + y^2)$?

Does there exist bivariate polynomials $p$ and $q$ such that $p(x,y)^2 = q(x, y)^2 ( x^2 + y^2)$ for all real $x$ and $y$?
1
vote
0answers
78 views

Solving an 8th degree polynomial

I know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...
6
votes
1answer
95 views

finite field extension problem

Maybe somebody knows how to proove the following algebraic theorem: $C \subset U$ is a field extension and $N \subset U$ so, that all $x \in N$ are algebraic over $C$ and $C[N]=\left\lbrace ...
4
votes
4answers
98 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 ...
1
vote
1answer
56 views

Irreducibility of $\frac{X^{p^n}-1}{X^{p^{n-1}}-1}$.

Exercise: Let $p$ be a prime number. Then, the polynomial \begin{equation} \frac{X^{p^n}-1}{X^{p^{n-1}}-1} \end{equation} is irreducible over $\mathbb Z[X]$, for any integer $n \geq 1$. I'm able to ...
6
votes
1answer
196 views

Why is the polynomial $S(\vec{x})$ with coefficients obeying a constraint homogeneous?

I have recently been working on a problem to prove that a particular polynomial is in fact homogeneous. Although I have found out that this is true, I am curious to see whether there might be a deeper ...
0
votes
1answer
48 views

Solutions of $x^d=1$ in a finite field

Let's consider the polynomial $x^d-1$. Theory tells us that it can have at most $d$ roots in (any extension of) a given field. Here's my problem: let $A$ be the vector space spanned by ...
0
votes
1answer
27 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 $2$ ...
3
votes
3answers
70 views

Roots of unity modulo $p$

Let $f(X)$ be the minimal polynomial of something like $\zeta + \frac{1}{\zeta}$, where $\zeta$ is a primitive $p$-th root of unity for some prime $p > 2$. I'd like to show that $f(X) \equiv ...
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 ...
1
vote
1answer
32 views

When is the Frobenius endomorphims an isomorphism?

I did this problem, but now I'm left with more questions! Suppose $f(x)$ is a monic irreducible polynomial of degree $3$ over $GF(2)$. Prove that if $a$ is a root of $f$ in an extension of $GF(2)$, ...
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 ...
2
votes
2answers
54 views

Show that $P(X) -X$ divides $P(P(X))-X$

Let $P$ be a polynomial in $R[X]$. Then show that $P(X) -X$ divides $P(P(X))-X$
1
vote
1answer
50 views

Why are $(X_1), (X_1,X_2), \ldots$ prime ideals?

I was looking at the proof of the dimension of the polynomial ring $R[X_1,\ldots,X_n]$ and I had a question: Why are $(X_1), (X_1,X_2), (X_1,X_2,X_3),\ldots, (X_1,\ldots,X_n)$ prime ideals in this ...
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
0answers
47 views

How to prove how many ireducible polynomials are in a polynomial ring over a finite field.

Can someone help me out with some problem concerning the cardinaliy of ireducible polynomials in $(\mathbb Z/p\mathbb Z)[x]$? Here first why the problem appeared, I want to prove that ...
1
vote
1answer
32 views

If $[E:F]$ is finite and $\alpha \in E$ then there is an irr. polynomial in $F[x]$ with root $\alpha$

I'm studying for an exam and encountered a confusing proof of the following fact in my notes: Let $[E:F]$ be finite and $\alpha \in E$ then there is an irreducible polynomial $p(x) \in F[x]$ with ...
0
votes
3answers
28 views

For an irreducible polynomial $p$ and root $\alpha$, $[F[\alpha]:F]$ = degree of $p$

I'm studying for an exam, and I couldn't find the proof for the following theorem in my notes: For $p(x) \in F[x]$ irreducible and $\alpha$ a root of $p$ in some extension field, $[F[\alpha]:F] =$ ...
0
votes
1answer
35 views

Problems with a ring isomorphism

Let $k$ be a field and consider $a=(a_0,\ldots,a_n)\in k^{n+1}$ with $a_0\neq0$. Now $\rho(a)=\left(\{a_iT_j-a_jT_i\;:\; 0\le i<j\le n\}\right)$ is an homogeneous ideal of $k[T_0,\ldots,T_n]$ and I ...
0
votes
1answer
37 views

Algorithms for factoring multivariate polynomials

I am wondering if there are any algorithms to factor polynomials in multiple variables, when you know that the factors are other polynomials with rational or integer coefficients. I know you have the ...
1
vote
0answers
48 views

Solving for Coefficients of Multivariable Polynomial from Expanded Form

I have 3 single variable polynomials in x, y, and z that have been multiplied together (expanded). I have the numerical coefficients of the expanded polynomial and want to solve for the a, b, and c ...
3
votes
3answers
91 views

Homogenous polynomial over finite field having only trivial zero

Is there a way to construct homogenous polynomials of small degree over a certain finite field having only trivial zero? For instance, the polynomial $f (a, b, c) = a^3 + b^3 + c^3 - 3abc - 3a^2b - ...
2
votes
1answer
77 views

Monic irreducible polynomial over an integral domain

These days, I have some basic problem in abstract algebra. I know that in any integral domain, any prime element must be an irreducible element. Moreover, if $A$ is a UFD, then an element $a \in A$ is ...