# Questions tagged [galois-theory]

Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood.

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### Is Galois theory necessary (in a basic graduate algebra course)?

By definition, a basic graduate algebra course in a U.S. (or similar) university with a Ph.D. program in mathematics lasts part or all of an academic year and is taken by first (sometimes second) ...
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### Does one real radical root imply they all are?

Is there an example of an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with a real root expressible in terms of real radicals and another real root not expressible in terms of real radicals?
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### Which small finite simple groups are not yet known to be Galois groups over Q?

The subject line pretty much says it all. To expand just a little bit: 1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known ...
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### Grothendieck says: points are not mere points, but carry Galois group actions

Apologies in advance if this question is too elementary for MO. I didn't find an explanation of these ideas in any algebraic geometry books (I don't know French). The following is an excerpt from ...
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### Fields with trivial automorphism group

Is there a nice characterization of fields whose automorphism group is trivial? Here are the facts I know. Every prime field has trivial automorphism group. Suppose L is a separable finite extension ...
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### Abel and Galois (and Arnold)

Question Is there a connection between Abel and Galois theories of polynomial equations? Recall that for every polynomial $p(x)\in \mathbb{Q}[x]$ (say, without the free coefficient), Abel considered ...
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### transcendental Galois theory

Suppose we define an arbitrary field extension $K/F$ to be Galois if, for all subextensions $L$ of $K/F$, we have $K^{\operatorname{Aut}(K/L)} = L$. In words: for any element $x$ of $K \setminus L$, ...
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### Is my field algebraically closed?

For a field $L$, let $\widetilde L$ be the splitting field of all irreducible polynomials over $L$ having prime-power degree. Question: Do we have $\widetilde{\mathbf Q}=\overline{\mathbf Q}$? My ...
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### Does any cubic polynomial become reducible through composition with some quadratic?

What I mean to ask is this: given an irreducible cubic polynomial $P(X)\in \mathbb{Z}[X]$ is there always a quadratic $Q(X)\in \mathbb{Z}[X]$ such that $P(Q)$ is reducible (as a polynomial, and then ...
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### Fundamental groups of topoi

Just yesterday I heard of the notion of a fundamental group of a topos, so I looked it up on the nLab, where the following nice definition is given: If $T$ is a Grothendieck topos arising as category ...
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### $A_5$-extension of number fields unramified everywhere

So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...
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### Grothendieck's "La longue Marche à travers la théorie de Galois"

It seems that Grothendieck's familly has given permission for the distribution of his unpublished works, so I hope it is ok to ask this. Is there any way to obtain a copy (online or not) of "La ...
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### Solving the cubic by "radicals" in characteristics 2 and 3

This question has no justification other than a bit of fun. We all know that the cubic is solvable by "radicals" ($\root2\of{}$ and $\root3\of{}$) in characteristics $\neq2,3$. The formula was ...
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### Galois Group as a Sheaf

I've noticed that the Galois groups associated to Galois field extensions $L$ of a given field $K$ seem remarkably like a sheaf, with the field extensions taking the place of open set, and the Galois ...
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### Is there a Galois correspondence for ring extensions?

Given an ring extension of a (commutative with unit) ring, Is it possible to give a "good" notion of "degree of the extension"?. By "good", I am thinking in a degree which allow us, for instance, to ...
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### Why do some uniform polyhedra have a "conjugate" partner?

While browsing through a list of uniform polynohedra, I noticed that the square of the circumradius $R_m$ of the small snub icosicosidodecahedron ($U_{32}$) with unit edge lengths is, R_{32}^2 =\...
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### How are motives related to anabelian geometry and Galois-Teichmuller theory?

In Recoltes et Semailles, Grothendieck remarks that the theory of motives is related to anabelian geometry and Galois-Teichmuller theory. My understanding of these subjects is not very solid at this ...
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### Grothendieck's Galois theory without finiteness hypotheses

This is motivated by the discussion here. The usual definition of the etale fundamental group (as in SGA 1) gives automatic profiniteness: Grothendieck's formulation of Galois theory states that any ...
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### Anabelian geometry study materials?

I want to study anabelian geometry, but unfortunately I'm having difficulties in finding some materials about it. If you could offer me some books/papers/articles I would be glad.
### Galois Group of $x^n-2$
Let $n \in \mathbb{N}$, then the order of the Galois Group of $x^n-2$ coincide with $n \phi(n)$ for $n\in \{ 1 , \dots , 36 \}$ except for $n=\{ 8, 16, 24, 32 \}$ where this order is \$\frac{ n \phi (n)...