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
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 ...
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, ...
1
vote
0answers
24 views

The existence of Pisot numbers in any real number field

Wikipedia claims that, given a real algebraic number field $K$ of degree $n$, there is an algebraic integer $r \in K$ of degree $n$ such that $r>1$, but every conjugate of $r$ has modulus $<1$ ...
8
votes
2answers
195 views

Checking irreducibility

I have the polynomial $f(X)=X^{2n}-2X^{n}+1-p$ where $p$ is a prime number and $n\in\mathbb{N}$. I want to check whether it is irreducible or not over $\mathbb{Q}[X]$. If $2^{2}\nmid1-p$ then $f(X)$ ...
2
votes
1answer
64 views

Field not closed under complex conjugation

What is an example of an algebraic field not closed under complex conjugation? In all subfields of $\mathbb C$ I think of, complex conjugation is a transposition. I think I understand that it is ...
1
vote
1answer
52 views

Galois group of an irreducible cubic polynomial without using the discriminant

Let $f\in\mathbb Q[x]$ be irreducible of degree $3$. Since the Galois group $G$ of $f$ is a transitive subgroup of $S_3$, it is either $S_3$ or $A_3$. Those two possibilities are easily ...
5
votes
4answers
277 views

Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
2
votes
0answers
54 views

Proof that $p(z)^2=a^2$ always has a nonreal solution.

Let $p(z)$ be a nonconstant integer polynomial of degree $n$ such that $p(0)=0$ and let $a$ be a nonzero real number. It seems that $$p(z)^2=a^2$$ Always has a nonreal solution (in $z$) if ...
1
vote
1answer
105 views

Galois group of a quartic

Let $x^4+ax^2+b$ in $K[x]$ (with char $K\neq $2) be irreducible with Galois group $G$. (a) If $b$ is a square in $K$, then $G = \mathbb{Z}_2\times\mathbb{Z}_2$. (b) If $b$ is not a square in $K$ ...
1
vote
2answers
51 views

Why $S_4$ has no transitive subgroups of order 6?

I know that every transitive subgroup of $S_4$ have to be order divisible by 4, but i should solve this with Galois Theory. I think this theorem can be usefull: Theorem 4.2. Let K be afield and f in ...
2
votes
1answer
37 views

Help with Polynomial Roots Problem

Let's consider the case of two variables, $p\in\mathbb{R}[x,y]$. Suppose I want to find when there is $c\in\mathbb{R}$ such that $$p(x,x)+p(x,c-x)-p(c-x,x)-p(c-x,c-x)=0 \textbf{ for all } ...
1
vote
1answer
28 views

Find a polynomial whose splitting field is $\mathbb{Q}[\alpha,i]$

Let $f(x)=x^{3}-3x+1$ and let $\alpha$ be a root in $f$. i) Show that the polynomial $f$ is irreducible in $\mathbb{Q}[x]$. ii) Show $\alpha^{2}-2$ is a root of $f$ as well, and show that all roots ...
0
votes
1answer
46 views

Solving equations in cosines

Consider the equation $9x^4+27x^3-33x^2-153x-101=0$. Its Galois Group is $C_4$. That means by Kronecker-Weber theorem that it is solvable in cosines. How can you find this solution? For example, ...
4
votes
1answer
106 views

Determine Galois groups of polynomials

Determine the Galois groups of the following polynomials over Q. $f(x)=x^3−3x+1$. $g(x)=x^4+3x+3$. $h(x)=x^5+8x+12$. I have no way to find their Galois groups. I only obtained that since they are ...
3
votes
2answers
44 views

minimal polynomial of roots of irreducible polynomial

Let $f\in \mathbb{Z}[x]$ be irreducible over $\mathbb{Q}[x]$, the highest degree coefficient of $f$ is $1$. Let $\omega\in \mathbb{C}$ such that $f(\omega)=0$. Can we obtain that the minimal ...
0
votes
1answer
27 views

Defining and describing a field extension being normal

My notes by Jens Carsten Jantzen (department of Mathematics at the University of Aarhus) defines a field extension as normal if: $N\supset K$ is a normal field extension if for each $\alpha\in N$ ...
4
votes
2answers
61 views

Given $f \in \mathbb{Q}[x]$ irreducible. How many and which roots of $f$ are contained in $\mathbb{Q}[x]/(f)$?

It is a fact that struggle me for a while. When working with irreducible polynomial over $\mathbb{Q}$ it is natural to build the extension ${\mathbb{Q}[x]}/{(f)} $ in which "lives " one root of the ...
5
votes
1answer
98 views

show that $X^4-4X^2-21$ is solvable by radicals

show that $$X^4-4X^2-21\in\mathbb{Q}[X]$$ is solvable by radicals. $\mathrm{Def}$: Let $f(X)\in K[X]$ and let $\Sigma$ be a splitting field for $f(X)$ over $K$. We say $f(X)$ is solvable by ...
7
votes
1answer
82 views

Does $f=g_1^{n_1}\cdots g_k^{n_k}$ imply $Gal(f)=Gal(g_1)\times\cdots\times Gal(g_k)$?

Let $f\in \mathbb{Z}[x]$ and $f=g_1^{n_1}\cdots g_k^{n_k}$ where $g_1,\cdots, g_k$ are distinct irreducible polynomials over $\mathbb{Q}$. Whether does it hold $Gal(f)=Gal(g_1)\times\cdots\times ...
4
votes
2answers
82 views

Are the elementary symmetric polynomials “unique”?

The elementary symmetric polynomials are interesting in that they generate the set of symmetric polynomials, in the sense that every symmetric polynomial is some polynomial applied to the elementary ...
3
votes
1answer
31 views

Which of the following polynomials are separable?

Which of the following polynomials are separable? a)$\;t^4-8t^2+16\in\mathbb{Q}[t]$ b)$\;t^{17}-t\in\ \mathbb{F}_{17}[t]$ c)$\;t^{17}-X^{17}\in \mathbb{F}_{17}(X)[t]$ a) My first idea to use the ...
1
vote
1answer
70 views

$\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$

I want to prove: $\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$. Is there any direct way to prove? I have computed that the splitting field of $x^7-12$ ...
0
votes
2answers
69 views

Intersection of splitting fields of two polynomials

Let $f,g\in \mathbb{Z}[x]$, $(f,g)=h\in \mathbb{Z}[x]$ (if this is not true, can we construct $h$ in another way?). Let $K_1,K_2$ be the splitting fields of $f,g$ over $\mathbb{Q}$ respectively. Is ...
1
vote
3answers
170 views

Galois group of an irreducible polynomial

Find the Galois group of the polynomial $x^5-9x+3$ over $\mathbb{Q}$. since $3$ cannot divide $a_5$, $3$ divide other coefficients, $3^2$ cannot divide $a_0$, we see that the polynomial is ...
4
votes
1answer
73 views

Solution of $Ax^5+Bx^3=C$

I have to find the positive solution of the type $Ax^5+Bx^3=C (A,B,C>0)$. It is well known that a polynomial of degree greater than $4$ does not admit an expression for the roots but I hope :D In ...
0
votes
1answer
36 views

Does the succesion of two radical extensions yield a radical extension only in the obvious case?

Answering that recent stackoverflow question, I encountered the following related problem : Let $n,m,p\geq 2$ be integers, and let $K$ be a subfield of $\mathbb C$ containing all $nmp$-th roots of ...
12
votes
1answer
211 views

Semialgebraic conditions that convey properties of Galois group

Let $f \in \mathbb{Z}[x]$ be a polynomial of degree $n$ with integer coefficients and let $G_f$ be the Galois group of $f$ over $\mathbb{Q}$. I am trying to collect results that convey some ...
1
vote
1answer
90 views

$F$ field, $\alpha$ separable on $F$. Is $F(\alpha)$ a separable extension of $F$?

Let $F$ be a field, and let $\alpha$ be algebraic and separable over $F$. Is $F(\alpha)$ a separable extension of $F$? By "$\alpha$ is separable" I mean that its minimum polynomial over $F$ is ...
4
votes
1answer
278 views

Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$

So I want to show that $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ and determine its Galois group. My thoughts are as follows: Define $\alpha := \sqrt{2+\sqrt{2}}$. Then it is ...
1
vote
1answer
90 views

On irreducible polynomial over normal extension

Let $L/K$ be a normal extension and a irreducible polynomial $f(X) \in K[X]$. Prove that, if $f$ is reducible over $L$ then $f$ is factored into product of irreducible factors with same degree. ...
0
votes
2answers
62 views

Cubic with repeated roots has a linear factor

If $f$ is a cubic polynomial with a repeated root over a field then $f$ has a linear factor. I think that is true in perfect fields but I don't know how to prove it.
1
vote
1answer
48 views

A question about cubic roots of rational numbers

I'm trying to understand if, given $K$ a cubic cyclic extension of $\mathbb{Q}(\zeta_3)$, where $\zeta_3$ is a third primitive root of unity, it always exists $b \in \mathbb{Q}$ such that $\sqrt[3]{b} ...
1
vote
2answers
84 views

On polynomial of prime degree.

Let $K$ be a field, $f(X)\in K[X]$ be a polynomial of prime degree. Assume that for all extension $L$ of $K$, if $f$ has roots in $L$ then $f$ splits over $L$. Prove that either $f$ is irreducible ...
2
votes
0answers
58 views

Galois group of $x^5-2px+p$ over $\mathbb{Q}$ with $p$ prime

I proved that $x^5-2x+p$ is irreducible in $\mathbb{Z}$ so in $\mathbb{Q}$ by Gauss Lemma I need (and I still can't) to prove that $x^5-2x+p$ has exactly three real roots and conclude that the Galois ...
-1
votes
2answers
99 views

Generalizations of the quadratic formula [duplicate]

The quadratic formula can be used to find the roots of any quadratic polynomial of the form $ax^2 + bx + c$: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ The derivation is simple enough and uses a ...
2
votes
2answers
96 views

Generator Polynomial and Minimum Distance

Given a generator polynomial, how do I calculate minimum distance for the code. I am working in GF(2). A particular case of interest is the cyclic code of length $9$ generated by $$ ...
1
vote
1answer
77 views

Automorphisms and Splitting Fields

Note: This question comes from a non-examined question sheet from an undergrad maths course. I want to find the splitting fields of the following polynomials: $x^3-1$ over $\mathbb{Q}$ $x^3-2$ over ...
0
votes
2answers
64 views

Irreducibility of Polynomial in $\mathbb{Q}$

How can we show that, if $a>1$ is the product of distinct primes, then $x^n-a$ is irreducible in $\mathbb{Q}$ for all $n \geq 2$ and that it has no repeated roots in any extension of $\mathbb{Q}$? ...
3
votes
0answers
73 views

Question about Wantzel's proof of the necessary condition for compass/straightedge constructibility

I'm trying to understand Wantzel's original proof of the necessary condition for constructibility with a straightedge and compass. It's expressed in terms of polynomials rather than field extensions. ...
8
votes
4answers
601 views

showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$

I studied the cyclotomic extension using Fraleigh's text. To prove that Galois group of the $n$th cyclotomic extension has order $\phi(n)$( $\phi$ is the Euler's phi function.), the writer assumed, ...
1
vote
0answers
97 views

Determine generator over $GF(2^4)$

Working in $GF(2^4$) Field generated modulus $x^4+x^3+x^2+x+1$. Find a generator of $F$. What I have figured out so far - $16$ polynomials to consider. If $b$ is generator then start with $b = x + ...
9
votes
1answer
159 views

Why do you need to introduce complex numbers in order to solve cubics by radicals?

Historically, complex numbers were first introduced (by Bombelli) in order to provide a method for solving the cubic by radicals. As far as I can tell, previous solutions of the cubic by radicals ...
4
votes
1answer
121 views

Is the real root of $x^4+3 x-3$ constructible?

I don't know how to solve the following: Let $\alpha$ be a real root of $f(x)=x^4+3x-3\in Q[x]$. Is $\alpha$ a constructible number? Any help is welcome.
3
votes
0answers
71 views

How does Dedekind's Theorem work for a prime dividing the discriminant of a number field?

Let $f \in \mathbb{Z}[x]$ be an irreducible monic polynomial, let $N$ be its splitting field, and let $G$ be the Galois group of the extension $N/\mathbb{Q}$. Let $p$ be a prime dividing the ...
1
vote
0answers
57 views

Minimal Polynomials of numbers from trigonometry

What is the minimal polynomial over $\Bbb Q[x]$ of $$\alpha[k] = n\cdot ...
3
votes
2answers
111 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 ...
4
votes
2answers
78 views

Is there a splitting field for multivariate polynomials over $\mathbb{Q}$?

Let $f(X,Y) = X^2 + Y^2 - c$. Does there exist a field $F$ such that $f(X,Y) = \prod_{i=1}^n(a_1^i X + a_2^i Y + a_3^i)$, where $a^i$'s are in $F$?
2
votes
0answers
92 views

Can one trisect the angle $\theta=\arccos(-12/17)$?

I think this is not possible, and below is my proof. Proof: Using the angle tripling formula: $$\cos(3\theta) = 4\cos^3 (\theta)−3\cos(\theta) \ \to \ \cos(\theta) = 4\cos^3 (\theta/3) ...
2
votes
2answers
222 views

Prove that the angle $\theta=\arccos(-12/17)$ is constructible using ruler and compass.

Could I just do this? Proof: if we want to show $\arccos\left(\frac{-12}{15}\right)$ is constructible, can't I just say, take $x_0=\cos(\theta)=-\frac{12}{17}$ implies $17x_0+12=0$ which says that ...