This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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2
votes
1answer
22 views

Deciding if $i \in \mathbb{Q}(\alpha)$ for the root $\alpha$ of a certain polynomial

Consider the field $\mathbb{Q}(\alpha)$, where $\alpha$ is one of the (complex) roots of the polynomial $f(x) := x^3 + x + 1 \in \mathbb{Q}[x]$. I now want to find out if $i \in \mathbb{Q}(\alpha)$ ...
4
votes
2answers
47 views

Showing that $(x^2 - 2)(x^2 - 3)(x^2 - 6)$ has a root in $\mathbb{F}_p$

Let $p$ be a prime number, $K = \mathbb{F}_p$ the field with $p$ elements, and $f = (x^2 - 2)(x^2 - 3)(x^2 - 6) \in K[x]$. I now want to show that $f$ has a root in $K$. I know that to show the ...
1
vote
1answer
76 views

Synthetic division for: $\frac{60 x^{3}+43x^{2}-34x-24}{3x+2}$ [duplicate]

If I have a polynomial to which the solutions are integers, in this case, I know how to perform the synthetic division. Also, I know how to perform the present division using long division. But I don'...
0
votes
0answers
19 views

Applications of discriminant and resultant

In Algebra, discriminant (of a polynomial) and resultant (of two polynomials) are usually introduced with focus towards multiple or common roots of polynomial(s). Beyond this thread - finding common ...
-1
votes
1answer
38 views

$x^3-3ax+b$ has $3$ real roots and $a$ and $b$ are real. The roots are $m < n < p$.

$x^3-3ax+b$ has $3$ real roots and $a$ and $b$ are real. The roots are $m < n < p$. Then show that: $m < - \sqrt {|a|} < n < \sqrt {|a|} < p$. where $a>0$ By Vieta: $m+n+p=0$, $...
1
vote
1answer
59 views

Can someone suggest a way to simplify $x_1(y_1 - x^Ty) + x_1(w_1 - x^Tw)^2 - x_1^2(w_1 - x^Tw)^2 + x_1x_2(w_1 - x^Tw)^2$

Let $x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, $y = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$, $w =\begin{bmatrix} w_1 \\ w_2 \end{bmatrix}$ I have the following vector: $V = \begin{bmatrix} ...
0
votes
0answers
13 views

How to determine the truncation error with products and quotients

If I have an equation given by $$\displaystyle Y = \frac{a^2}{d^2}\frac{(1-c^2\frac{c}{a})}{(1-b^2)}$$ and I expand $a,b,c,d$ in a Taylor series, where $a$ is truncated at the $A^{th}$ order, $b$ is ...
-2
votes
2answers
37 views

$(f_1(x), f_2(x), …, x-a) = (f_1(a), …, f_r(a), x-a)$ with $a \in R$, $R$ is a commutative ring, $f_i(x) \in R[x]$ [closed]

Let $R$ be a commutative ring, $a \in R$, and $\forall i = 1, ...,r \ \ f_i(x) \in R[x]$. Prove the equality of ideals $(f_1(x), ..., f_r(x), x-a ) = (f_1(a), ...f_r(a), x-a)$. That is, $\forall ...
0
votes
1answer
30 views

Print LIST of polynomials in Magma in one line instead of multiline

Suppose I have list of two polynomials and want to print it. I will get multi line output: ...
1
vote
2answers
41 views

Cubic Polynomial With an Extra Variable

I was trying so desperately to find the value of $c$. Let $$P(x) = 2x^3+2x^2-2cx+4$$ $x+2$ is the factor of $P(x)$. So I started factoring it, going $x+2(2x^2-2x...$ and then got stuck at factoring $...
0
votes
1answer
63 views

Number Theory and p-Remainder Numbers

In order to submit the problem, here it comes the definition we are interested in. Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$ and some natural $p > 1$, we will designate a p-remainder ...
0
votes
1answer
32 views

How to prove that $h''(x)$ has at most one zero on $(0,1)$.

$h(x)=1-\sum_{i=1}^{k-1}x^i+a_kx^k+\sum_{i=k+1}^\infty x^i$, where $|a_k|\le1$, is the power series of an analytic function. Prove that $h''(x)$ has at most one zero on $(0,1)$.
0
votes
0answers
16 views

Closed-form formula for system of two bivariate quadratic polynomials

Given a system of two bivariate quadratic polynomials: \begin{eqnarray} a_0 + a_1 x + a_2 y + a_3 xy+a_4 x^2 + a_5 y^2 &= 0 \\ b_0 + b_1 x + b_2 y + b_3 xy+b_4 x^2 + b_5 y^2 &= 0 \end{...
4
votes
2answers
139 views

Irreducible polynomial over $\mathbb{Q}$ implies polynomial is irreducible over $\mathbb{Z}$

Let $f(x) \in \mathbb{Z}[x]$ be a polynomial of degree $\geq 2$. Then choose correct a) if $f(x)$ is irreducible in $ \mathbb{Z}[x] $ then it is irreducible in $ \mathbb{Q}[x] $. b) if $f(x)$ is ...
0
votes
0answers
24 views

Let $(f_1,\ldots,f_r)\subset k[x_1,\ldots,x_n],(g_1,\ldots,g_s)\subset k[y_1,\ldots,y_m]$ be radical ideals. Then their sum is radical. [duplicate]

Let $\mathfrak{a}:=(f_1,\ldots,f_r)\subset k[x_1,\ldots,x_n],\mathfrak{b}:=(g_1,\ldots,g_s)\subset k[y_1,\ldots,y_m]$ be radical ideals. Then I wish to prove that $\mathfrak{c}:=(f_1,\ldots,f_r,g_1,\...
0
votes
0answers
70 views

Number Theory and p-Progressive Numbers

Before proposing the problem itself, it shall be profitable to define $b_{p}(k) = k^{p}$. In other words, the sequence $b_{p}(k)$ is an arithmetic progression of order p. For the sake of our purposes, ...
1
vote
3answers
67 views

$x=p^{k/5}+p^{l/5}+p^{m/5}+p^{n/5}$ is a root of a quintic equation for any $p \in \mathbb{Q}$ and $k,l,m,n \in \mathbb{Z}$?

I first derived a special case of $k=1,~l=2,~n=3,~m=4$: $$x=p^{1/5}+p^{2/5}+p^{3/5}+p^{4/5}$$ $$x^5-10 p x^3-10p(p+1)x^2-5p(p^2+p+1)x-p(p^3 + p^2 + p + 1)=0$$ More general case works (even though ...
7
votes
0answers
190 views

When is the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)+1$ reducible in $\mathbb{Z}[x]$?

This post is inspired by Prove that the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)-1$ is irreducible in $\mathbb{Z}[x]$.. (A) Find all positive integers $n$ and integers $a_1,a_2,\...
1
vote
1answer
81 views

Prove that the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)-1$ is irreducible in $\mathbb{Z}[x]$.

Let $n>1$ be an integer. For $a_1,a_2,\ldots,a_n\in\mathbb{Z}$ with $a_1< a_2< a_3 < \dots < a_n$, prove that the polynomial $$f(x)=(x-a_1)(x-a_2)\cdots(x-a_n)-1\,.$$ is irreducible in ...
2
votes
1answer
53 views

Real roots of a polynomial with all its coefficients equal to 1 or -1

Dears, I am interesting to know how many (real) roots can have the polynomial $p(x):=1+a_{1}x+\ldots+a_{n}x^{n}$ in the interval $(-1,0)$, where $a_{i}\in\{-1,1\}$ for all $i=1,\ldots,n$. I think ...
1
vote
1answer
87 views

Dependence of algebraic elements in a finite field

Lets work over the finite field $\mathbb{F}_p$ for a prime $p$. Consider a monic irreducible polynomial $f(X)=X^3+aX^2+bX+c$ in $\mathbb{F}_p[X]$. Let $x$ be a root of $f(x)=0$ (say, in the closure of ...
0
votes
1answer
39 views

Find a,b,c to match the linear transformation matrix?

P.S. Sorry for my bad explanation of the task, it was really hard to translate this into meaningful english For the given linear-transformation $A$ find all possible combinations of a,b,c for which ...
1
vote
1answer
46 views

Is L a linear transformation?

I have to prove is L is a linear transformation on the field $P_3(R)$, if it is then I'd have to find the matrix of the linear transformation from the standard base vectors $p(1),p(x),p(x^2),p(x^3)$. ...
0
votes
1answer
60 views

What are all polynomials $p(x)$ such that $p(q(x))=q(p(x))$ for every polynomial $q(x)$?

I assume that $p(x)$ and $q(x)$ are both real polynomials. If I let $q(x)=c$, (a constant) then $p(q(x)) = p(c) = q(p(x)) = c\ \forall c$. So $p(x)=x\ \forall x$. Is this operation valid and how ...
0
votes
0answers
15 views

Divisor Function over a Quadratic

The divisor function is defined as $\sigma_1(n)=\sigma(n)=\sum_{d\mid n}d$. Consider the divisor function over a quadratic $$f(x)=\sigma(a x^2+bx+c)$$ Where $a,b,c \in \mathbb{Z}$ (note we allow $a, b$...
0
votes
0answers
38 views

Solution to a non linear system of equations

Does there exist an interval I such that this system of equations has no solution? ...
5
votes
1answer
27 views

Derivative of a characteristic polynomial at an eigenvalue

Let $p(\lambda)$ be the characteristic polynomial of an $n\times n$ matrix $A$. We know that the roots of $p(\lambda)$ are the eigenvalues of $A$, hence the sum of the roots of the polynomial (taking ...
0
votes
0answers
18 views

What are the fixed points of $\alpha^n-\mu_j$ for a fixed $j$?

Let us consider the polynomial ring $\Bbb C[x_1,...,x_s]$ and $\alpha(x_i)= x_i + \mu_i$ where $\mu_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x_1,...,x_s])$. What are ...
6
votes
3answers
71 views

Prove that for any polynomial $P(x)$ there exist polynomials $F(x)$ and $G(x)$ such that $F\left(G(x) \right)-G\left(F(x) \right)=P(x)$

Prove that for any polynomial $P(x)$ there exist polynomials $F(x)$ and $G(x)$ such that $\forall x \in \mathbb R:$ $$F\left(G(x) \right)-G\left(F(x) \right)=P(x)$$ My work so far: Let $G(x)=x+1$...
2
votes
1answer
47 views

Is it possible to express the inverse of a polynomial as a series?

Is it possible to express the multiplicative inverse of a polynomial in descending powers of n i.e. \begin{equation} \frac{1}{\left[\sum_{k=0}^\infty a_kt^{n-2k}\right]^2} \end{equation} as a series ...
1
vote
0answers
44 views

Is there a special name for polynomials related by Möbius tranformation of the variable?

If we take a general polynomial with complex coefficients: $$C_n z^n+C_{n-1}z^{n-1}+\dots+C_1z+C_0$$ We can apply a general Möbius tranformation to the variable: $$z=\frac{aw+b}{cw+d},~~~~a,b,c,d \...
0
votes
4answers
78 views

How do I find the solution(s) to my second-degree equation?

$$f(x) = x^2 - 3x$$ My attempt : $$ \begin{align} x^2-3x &= 4\\ x(x-3) &= 4\\ x-3 &= 4 \\ x &= 7\\ \end{align} $$ I managed to solve one part of this problem but that one part is ...
0
votes
0answers
76 views

Proof of Ramanujan's famous cubic identity

Ramanujan found that given a polynomial $y=x^3+ax^2+bx+c$, one can find $\sqrt[3]{u+x_1}+\sqrt[3]{u+x_2}+\sqrt[3]{u+x_3}=\sqrt[3]{3\sqrt[6]{d}+w}$ where $$d=\frac {4(a^2-3b)^3-(2a^3-9ab+27c)^2}{27}$$$$...
1
vote
3answers
56 views

How to evaluate GF(256) element

I wonder is there any easy way to evaluate elements of GF$(256)$: meaning that I would like to know what $\alpha^{32}$ or $\alpha^{200}$ is in polynomial form? I am assuming that the primitive ...
2
votes
1answer
39 views

Interpolation for $f(n),n\in\mathbb{Z}$: Does it converge?

Assume a function $f(n)$ which is defined for $n\in\mathbb{Z}$. For each period $[n,n+1]$ the function could be interpolated with a polynomial of degree $m$. The polynomials should be built in a way ...
0
votes
2answers
51 views

Meaning of Vector Space over $\mathbb{R}$ being a Subspace of $\mathbb{R^R}$

$\mathscr{P(\mathbb{R})}$ is the set of all polynomials with coefficients in $\mathbb{R}$. How are below sentences related and why? (1) $\mathscr{P(\mathbb{R})}$ is a vector space over $\mathbb{R}...
1
vote
0answers
14 views

Spectrum of Kernel - Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form $$K(m,n)= e^{-\frac{\beta}{4} (m+n+1)} \frac{2^{2+\frac{m+n}{2}}}{\sqrt{m! n!}} \frac{\sqrt{\pi}}{n-m} \left[ \frac{1}{\Gamma(-m/2)\Gamma(...
2
votes
0answers
50 views

Factoring $x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$ over $\mathbb{Q}$

For a quntic polynomial to be reducible to the following form over $\mathbb{Q}$: $$x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$$ We need to match the coefficients ($a=B$ obviously, so we ...
2
votes
5answers
100 views

Factorize a third degree polynomial

It's my first time posting here so I'm not used to describing my problem in mathematics. I'm currently trying to solve a problem which asks if a 3x3 matrix is diagonalizable, I know the method but ...
0
votes
1answer
29 views

Given a polynomial with integer coefficients and prime independent term, show that any root has absolute value greater than 1.

I was looking at exercises about algebraic structures, and in ring theory I stumbled upon this problem. Given $p$ a prime number and $f(x)=\pm p + a_{1}x+\cdots+x^{n} \in \mathbb{Z}[x]$ so that $\...
4
votes
1answer
197 views

How to decide if a polynomial is symmetric?

First, is the following: $$f=\frac{3}{5}(x_1^5 + x_2^5 + x_3^5 + x_4^5)-\frac{7}{12}(x_1^2x_2^2 - x_1^2x_3^2-x_1^2x_4^2-x_2^2x_3^2-x_2^2x_4^2-x_3^2x_4^2)$$ a symmetric polynomial? And, if yes, how do ...
2
votes
2answers
38 views

Are a uniformly random polynomial's roots are distributed uniformly in the field?

Assume we have a $\mathbb{F}_p$, where $p$ is a large prime (e.g. 128-bit value). We define all polynomials over the field, and pick a polynomial,$P(x)$, of degree $d$, where the polynomials' ...
1
vote
2answers
35 views

primitive polynomials and their factorisation

A polynomial with integer coefficients is called primitive if its coefficients are relatively prime. For example, $$3{x^2} + 7x + 9$$ is primitive while $$10{x^2} + 5x + 15$$ is not. (a) Prove that ...
3
votes
3answers
119 views

Brumer quintic polynomials - is there a general formula for the roots?

There exist a family of quintic polynomials, called Brumer's polynomials (or Kondo-Brumer), which have the form: $$x^5+(a-3)x^4+(-a+b+3)x^3+(a^2-a-1-2b)x^2+bx+a,~~~a,b \in \mathbb{Q}$$ According to ...
0
votes
3answers
31 views

How to find other basis of polynomials of degree three or less? [closed]

How can i find a basis of polynomials of degree three or less, which is other than $\{1,t,t^2,t^3\}$ ?
3
votes
3answers
54 views

construct polynomial from other polynomials

If I have a polynomial, P, with root $a$ and a polynomial, Q, with root $b$, is there a way to construct polynomial R such that $a+b$ is a root of R? Here's a concrete example. a = $\sqrt2$. $P(x) = ...
1
vote
1answer
72 views

Coefficients of a Polynomial Approximation in Minimax Sense

I am working on a Uniform Random Number Generator using a IEEE paper, and I got stuck with the coefficients for a Piecewise Polynomial Approximation using Horner's Rule : $$ y = ((C_d x + C_{d-1})x +...
2
votes
1answer
48 views

Does there exist an algebraic solvability algorithm?

I was ruminating over quintics and got curious about the following idea. Consider a quintic equation: $$ Q(x) a_0 + a_1 x + a_2 x^2 + ... a_5 x^5$$ Such that the solutions to $$ Q(x) = 0 $$ Are ...
4
votes
2answers
96 views

'Strange' trigonometric roots of $x^5-4x^4+2x^3+5x^2-2x-1$ - could someone explain?

This quintic equation has $5$ real roots: $$x^5-4x^4+2x^3+5x^2-2x-1=0 \tag{1}$$ The roots are, from left to right: $$x_1=\frac{\cos \frac{19}{22} \pi}{\cos \frac{1}{22} \pi}$$ $$x_2=\frac{\cos \...
3
votes
1answer
32 views

$p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $A \to p(A)$ surjective on $M(n,\mathbb R)$?

Let $p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $f: M(n,\mathbb R) \to M(n, \mathbb R)$ defined as $f(A)=p(A) , \forall A \in M(n,\mathbb R)$...