3
votes
0answers
29 views

Determining if a given equation is solvable given a set of ultra-radicals

So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots) AS WELL AS a set of inverses for some polynomials which are not solvable using ...
5
votes
0answers
42 views

Is there any relation between Galois solvability and integrability of hamiltonian systems?

Galois theory provides a method and formalism to study solutions of polynomial equations and solvability. Dynamical Hamiltonian systems have a somewhat similar concept of integrability. Since many ...
2
votes
3answers
96 views

Is a polynomial equation of degree $\ge 5$ not solvable by any way?

By Galois Theory an arbitrary polynomial equation of degree $\ge 5$ is not solvable using radicals, unlike the polynomial equation of second degree which is solvable by radicals (because of the ...
1
vote
0answers
23 views

series of subgroups of the solvable group $Gal(x^6-7) $.

I have to solve the following question: Let $p(x) = x^6-7 \in \mathbb{Q}[x]$, then $Gal(p(x)) $ is a solvable group (the splitting field of $p(x)$ is contained in a radical extension). Give a series ...
4
votes
1answer
46 views

A Galois Extension over a field of characteristic p

Hi! I've been having a bit of trouble with this question, namely the very last part. It's from a past paper for a Galois Theory course, in which I have an exam on Monday. Basically, I can show every ...
0
votes
1answer
24 views

A group of order less than $60$

I'm given a theorem (call it Theorem 1) which states that $G$ is solvable iff there exists a chain of normal subgroups $\{e\} = G_0 \unlhd \ldots \unlhd G_n = G$ such that $G_i / G_{i-1}$ is abelian ...
3
votes
1answer
39 views

Clarification of Abel-Ruffini theorem statement

From what I've studied, Abel-Ruffini theorem states that we can't find all the roots of some polynomials with degree above 5, using only radicals and arithmetic operations. How does it imply that we ...
0
votes
1answer
71 views

Prove that $F(\sqrt{\alpha}) = F(\sqrt{\beta})$

Let $F$ be a field of characteristic $\neq 2$. State and Prove a necessary and sufficient condition on $\alpha, \beta\in F$ so that $F(\sqrt{\alpha})=F(\sqrt{\beta})$.
1
vote
1answer
54 views

An alternative definition of a solvable group

I'm putting together an outline for a paper on Galois theory. There are a few equivalent definitions of a solvable group, and I need to make sure that the one I'd like to use works, or more ...
0
votes
1answer
42 views

A question on solvable groups.

Let's suppose $S$ is a group and can be written as $S= G_1\times G_2\times\cdots\times G_K$; here each $G_i$ is a group of prime power order. And $o(S)= n.$ Is this $S$ solvable group? If yes how will ...
7
votes
4answers
460 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 ...
0
votes
0answers
60 views

Express the exact root using radicals.

In Galois Theory, when a polynomial is not solvable by radicals, does that imply that the real solutions cannot be expressed exactly using the operators $+,-,\times,/$ and/or radicals?
2
votes
1answer
119 views

Show that the minimal polynomial of every element in $K=\mathbb{Q}(\zeta)$ is solvable by radicals, where $\zeta$ is a primitive 9th root of unity.

I have found the minimal polynomial if $\zeta$ over $\mathbb{Q}$ is $x^{6}+x^{3}+1$. $\mathbb{Q}(\zeta)\colon\mathbb{Q}$ is a normal and separable extension so ...
3
votes
1answer
131 views

Solvable by radicals or not?

Let $f=2x^5-5x^4+5\in \Bbb Q[x]$. Then, how to prove that it's not solvable by radicals? Since $f$ is solvable by radicals iff $Gal(f)$ is solvable and $Gal(f)\subset S_5$ (up to isomorphism), I ...
7
votes
2answers
212 views

Solvability by radicals of an equation of prime degree

For which prime $p$ the equation $x^p-p^px+p=0$ is solvable by radicals? I don't know how to solve this for primes $p\neq 2$, so any help is welcome.
24
votes
1answer
337 views

Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...