# Tagged Questions

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.

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 $i \in \mathbb{Q}(\alpha)$ ...
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 ...
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'...
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 ...
38 views

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 ...
37 views

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 ...
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 $(0,1)$.
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{...
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 ...
24 views

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 ...
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 ...
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 ...
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 ...
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 $p(1),p(x),p(x^2),p(x^3)$. ...
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 ...
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 $a, b$...
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? ...
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 ...
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 ...
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 $G(x)=x+1$...
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. $$\frac{1}{\left[\sum_{k=0}^\infty a_kt^{n-2k}\right]^2}$$ as a series ...
44 views

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 ...
### 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 ...