3
votes
2answers
69 views

Cubic polynomial - radical expression of roots

Let $f=X^3+X^2-2X-1$ be a polynomial with the three roots $x_1,x_2,x_3$ with $x_1=2\text{cos}(\frac{2 \pi}{7})$. We define $z:=(x_1-x_2)(x_1-x_3)(x_2-x_3)$. I want to find a radical expression for ...
2
votes
1answer
38 views

How to show that it holds $|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$ for any root of $X^n+\sum_{k=0}^{n-1}a_kX^k$?

Let $z\in\mathbb{C}$ be a root of the complex polynomial $$f=X^n+\sum_{k=0}^{n-1}a_kX^k$$ I want to show that it holds $$|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$$ Proof: For $s>1$, consider ...
1
vote
0answers
27 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 ...
1
vote
0answers
133 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 ...
1
vote
2answers
35 views

Separability of $f(x) = (x-1)^2(x-3)$ over $\mathbb{Q}$

This is an example in Ash, Basic Abstract Algebra, ch.3.4 page 73 at the bottom (or here on page 11). It states that $f(x) = (x-1)^2(x-3)$ over $\mathbb{Q}$ is separable. But, $f'(x) = ...
1
vote
1answer
62 views

Roots of unity over $\mathbb{Q}$

I want to show the following proposition from Algebra, Hungerford V.8.9. If $n > 2$ and $\xi$ is a primitive $n$th root of unity over $\mathbb{Q}$, then $[\mathbb{Q}(\xi + \xi^{-1}) : ...
1
vote
1answer
44 views

polynomial of degree 3 over set of rationals having only two rational zeros.

Does there exist a polynomial of degree 3 over the set of rationals which has only two rational zeros?
0
votes
0answers
59 views

Application of Hensel's lemma

Show that the polynomial $\Phi(x)=x^2 -2 \in O(\widehat{\Bbb Q_2})[x] $ has no root in $\widehat{\Bbb Q_2}$, even though $\bar\Phi(x)\in E(\widehat{\Bbb Q_2})[x]$ has a root in $E(\widehat{\Bbb Q_2}) ...
0
votes
0answers
38 views

roots of a polynom in a localization of a UFD

let $ {R} $ be a UFD, $ Q $ the localization of $ R $. I need to find all the roots in $ Q[i] $ of the polynom: $ f(x) = x^4 + \frac {4} {5+i}x^3 - \frac {6+10i} {2+3i}x^2 - \frac {12} {5+i}x + \frac ...
1
vote
1answer
58 views

Let $p$ be a prime in $\mathbb{Z}$, find all roots of $x^{p-1}-1$ in $\mathbb{Z}_p$.

Let $p$ be a prime in $\mathbb{Z}$. Find all roots of $x^{p-1}-1$ in $\mathbb{Z}_p$. Attempt at Solution I have to solve $x^{p-1}-1=0(\text{mod }p)$ for $x\in\mathbb{Z}_p$. This becomes ...
1
vote
1answer
59 views

Prove $f(x)=9x^2-5y^2-34$ has no integral roots

Prove $f(x)=9x^2-5y^2-34$ has no integral roots. I have tried working this mod 2, 3, 4, 5, and 17, and some random others, to no avail. It is for a graduate course, so I am thinking maybe I tried to ...
1
vote
1answer
61 views

Fields of polynomials . Proving that a belongs to k as a root

if $f(x)\in k[x]$, where $k$ is a field, then $a\in k$ is a root of $f(x)$ iff $x-a$ divides $f(x)$ in $k[x]$. My result ... If $a$ is a root of $f(x)=q(x)(x-q)$ and if we let $f(x)=q(x)(x-a)$,then ...
2
votes
2answers
94 views

$a+b\sqrt{2}$ not a root of monic polynomial over $\mathbb{Z}$

Consider $a+b\sqrt{2}$ for $a,b \in \mathbb{Q}-\mathbb{Z}$ . I need to show that it cannot be a root of any monic polynomial with coefficients in $\mathbb{Z}$
3
votes
2answers
116 views

Formula for roots of polynomials

For a quadratic polynomial there exists a formula for its roots. I read that similarly for polynomials of degree 3 and 4 there also exists such a formula but that no such formulas exist for ...
12
votes
2answers
119 views

Fully factored integer polynomials with constant differences

Given a degree $d$, it is possible to construct a pair $(F,\delta),$ where $F$ is a polynomial in $\mathbb{Z}[X]$ and $\delta$ a non-zero integer, such that $F(X)$ and $F(X)+\delta$ both split into ...
-4
votes
1answer
184 views

Root of a quadratic equation that has modulus $1$

Let us suppose $\alpha \in \mathbb C$ and $|\alpha|=1$ and $\alpha$ satisfies a monic quadratic equation. Then prove that $\alpha^{12} =1$. Show me the right way to solve this. Thanks in advance.
4
votes
2answers
90 views

Finding root using Hensel's Lemma

Hensel's Lemma calculates root of a polynomial $\in \mathbb{Z}_p[X]$ but is there any other significance to other branches of mathematics or outside mathematics? Why is finding root of ...
3
votes
2answers
129 views

How to find the roots of $f(x)=x^{2}+2x+2$ in $\mathbb{Z}_{3}$ ? in $\mathbb{Z}_{5}$ ? in $\mathbb{R}$?

Normally I just guess a root and then smash one out in high degree functions, or complete squares or any other number of mathemagical tricks, but my textbook has decided to break numbers on me and I ...
1
vote
1answer
262 views

Multiple roots of a polynomial in two variables

Let $F\in\mathbb{C}[X,Y]$ be an irreducible polynomial and $n\in \mathbb{N}$, $n\ge1$, $p_i\in\mathbb{C}[X]$ for $0\le i\le n$, such that $$F(X,Y)=\sum\limits_{i=0}^{n}p_i(X)Y^{n-i}.$$ Let ...
3
votes
3answers
193 views

How to find the roots of $x³-2$?

I'm trying to find the roots of $x^3 -2$, I know that one of the roots are $\sqrt[3] 2$ and $\sqrt[3] {2}e^{\frac{2\pi}{3}i}$ but I don't why. The first one is easy to find, but the another two roots? ...
-1
votes
1answer
134 views

Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ with exactly 4 distinct roots

Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ so that it has exactly 4 distinct roots and factorize it as product of irreducible factors. I'm really struggling in finding such polynomial, so ...
46
votes
5answers
1k views

Why are the solutions of polynomial equations so unconstrained over the quaternions?

An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb ...
1
vote
2answers
81 views

Normalization of a univariate polynomial with real algebraic coefficients

Consider a polynomial in one variable $x$ with irrational coefficients which are algebraic, i.e., they have a defining polynomial. As an example, take $p(x) = (x-3)(x-\sqrt{2}) = ...
2
votes
2answers
302 views

Proof existence of field extension of $\mathbb{F}_p$ containing the $r$-th primitive root of unity

I have to show the following: Let $p$ be a prime and $r \in \mathbb{N}$ with $\gcd(r,p)=1$. Prove the existence of a field extension $E$ of $\mathbb{F}_p$ which contains an $r$-th primitive root ...
4
votes
1answer
238 views

Number of real roots of a separable real polynomial doesn't change under small perturbations

Say we have a polynomial with real coefficients and no repeated roots. Knowing that the roots of a polynomial vary continuously in the coefficients (so long as we don't change the degree), it seems ...
2
votes
1answer
118 views

Defining Algebraic Numbers

An algebraic number is a number that is a root of a polynomial with rational coefficients. Any finite combination of rational numbers that can be combined with the usual four operations +, -, *, /, ...
5
votes
3answers
3k views

How to find roots of $X^5 - 1$?

How to find roots of $X^5 - 1$? (Or any polynomial of that form where $X$ has an odd power.)
3
votes
4answers
1k views

How to find a polynomial from a given root?

I was asked to find a polynomial with integer coefficients from a given root/solution. Lets say for example that the root is: $\sqrt{5} + \sqrt{7}$. How do I go about finding a polynomial that has ...
4
votes
1answer
178 views

Tropical Lifting Lemma (Puiseux Series)

The general result in tropical geometry is $K$ algebraically closed valued field $I$ ideal of $K[x_1, \cdots, x_n]$ $V(I) = \lbrace \bar{a}\in K^n: f(\bar{a})=0 \text{ for all } f \in I \rbrace$. ...
24
votes
3answers
772 views

Galois groups of polynomials and explicit equations for the roots

Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the ...
0
votes
3answers
242 views

Which field will contain all the roots of a polynomial over $GF(p)$

Given a a polynomial with coefficients in $GF(p)$ and degree $d$, will that polynomial always have $d$ roots in $GF(p^d)$? The text I am reading seems to be implying that this is true but I can't see ...
3
votes
1answer
551 views

Polynomials, Rouche's theorem and index of vector fields

In the proof of Rouche's theorem I saw in a book, there are two points I failed to understand, or failed to prove myself. (if you aren't familiar with the theorem, please try to look at the two ...