108
votes

### Why do we make such big deal about the 'unsolvability' of the quintic?

I think that a large part of the difficulty we have in understanding why this result is considered important is that it is psychologically difficult to put oneself into the shoes of mathematicians of ...

69
votes

Accepted

### Ideas for introducing Galois theory to advanced high school students

I have now twice taught Galois theory to advanced high school students at PROMYS. This is a six week course, meeting four times a week, for students who already are comfortable with proofs and, in ...

Community wiki

52
votes

### Why do we make such big deal about the 'unsolvability' of the quintic?

It is not a big deal anymore (edit: I mean in modern math, not for teaching undergraduate math courses) and has not been for a long time. It was important historically because of the math that came ...

45
votes

Accepted

### Grothendieck says: points are not mere points, but carry Galois group actions

Suppose $k$ is a field, not necessarily algebraically closed. $\text{Spec } k$ fails to behave like a point in many respects. Most basically, its "finite covers" (Specs of finite etale $k$-algebras) ...

35
votes

Accepted

### What is the dimension of the mathematical universe?

My co-authors and I introduced a notion of dimension for forcing
extensions in the following paper:
Hamkins, Joel David; Leibman, George; Löwe, Benedikt, Structural connections between a forcing ...

34
votes

### Why do we make such big deal about the 'unsolvability' of the quintic?

Perhaps it's not a big deal that's it's true; but it's a big deal that we can know and prove that it's true. I don't mean because the ability to prove that it's true implies the ability to do other, ...

33
votes

Accepted

### Does any cubic polynomial become reducible through composition with some quadratic?

You should refer to Lemma 10 (page-233) in this paper by Schinzel where he proves that for any polynomial $F(x)$ of degree $d$ we have a polynomial $G(x)$ of degree $d-1$ such that their ...

30
votes

Accepted

### How are motives related to anabelian geometry and Galois-Teichmuller theory?

Two clarifications:
For anabelian geometry, you should ask how much information about a variety is contained in the Galois action on its etale fundamental group.
While it's true that the motivic ...

28
votes

### Why do we make such big deal about the 'unsolvability' of the quintic?

Mathematically, there is nothing special about radicals: solving algebraic equations by radicals is only one example of mathematical problems among others that are self-contained, comprehensive, ...

26
votes

Accepted

### Abel and Galois (and Arnold)

The action of the monodromy group of $w(z)$ on the fiber $p^{-1}(a)$ for a non-critical value $a$ of $p$ (that is $|p^{-1}(a)|=\deg p$) is the same as the action of the Galois group of $p(x)+z$ over $\...

25
votes

### How are motives related to anabelian geometry and Galois-Teichmuller theory?

There's a very deep connection between motives and Grothendieck-Teichmüller theory but it isn't well-understood yet. I can't even frame it precisely in higher genus, but at least I can frame a ...

24
votes

Accepted

### Is there an algebraic formula for the eigenvalues of a symmetric $n\times n$ matrix?

If I calculated correctly, $$\begin{pmatrix}2 & 1 & 0 & 0 & 0\newline 1 & 3 & 1 & 0 & 0\newline 0 & 1 & 1 & 1 & 0 \newline 0 & 0 & 1 & 1 &...

24
votes

### Absolute Galois group, number theory and the Axiom of Choice

In the absence of the axiom of choice, it is still possible to define the "usual" algebraic closure of $\mathbb{Q}$ because you can just explicitly enumerate all polynomials with integer ...

22
votes

### Ideas for introducing Galois theory to advanced high school students

There is a nice book, specially written for high school students:
V. B. Alekseev, Abel's theorem in problems and solutions. Based on the lectures of V. I. Arnold (to high school students), and also ...

Community wiki

21
votes

### Does the category of (algebraically closed) fields of characteristic $p$ change when $p$ changes?

If you do not impose an algebraically closed condition, no two are equivalent. This basically follows from your (3). Namely, observe that
An extension is finite if and only if it has finitely many ...

20
votes

### Why do we make such big deal about the 'unsolvability' of the quintic?

To understand what is "so special about radicals that makes solutions of algebraic equations in terms of them so desirable" we should look at the problem not from the point of view of modern ...

19
votes

Accepted

### Grothendieck's "La longue Marche à travers la théorie de Galois"

All the manuscripts have been finally made avaible by Montpellier university.
You can find all of them here, avaible in pdf.
The items related to “La longue Marche" are:
La "Longue Marche" à ...

19
votes

Accepted

### Can the positive root of this polynomial be expressed elementarily?

I believe that the class of functions that Iosif Pinelis is interested in is what I would call exponential-logarithmic functions or EL functions; that is, they are the functions that can be expressed ...

18
votes

Accepted

### Classify all the fields with abelian absolute Galois group

Geyer in Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist, Satz 1.13 and the paragraph after that, gives a full characterization of which abelian ...

18
votes

Accepted

### An extension of the Galois theory of Grothendieck

The point of view where this title comes from is that Grothendieck's theorem can be seen as a characterization of toposes of the form $BG$ for $G$ a profinite group. It shows that some toposes can be ...

17
votes

Accepted

### Why do some uniform polyhedra have a "conjugate" partner?

There is nothing particularly mysterious here. Roughly, fix the lattice (incidence relations of all faces) and the edge lengths. Then there are several realizations of this structure, typically a ...

17
votes

Accepted

### Cyclic cubic extensions and Kummer theory

It's just the map
$$x \mapsto y = \frac{x}{x^{\sigma}},$$
where the corresponding degree three extension of $\mathbb{Q}$ is the degree three subfield of $k(y^{1/3})$. The point is that it is ...

17
votes

### Absolute Galois group, number theory and the Axiom of Choice

There would be no consequences, for two reasons:
As Timothy Chow points out, if we define $\overline{\mathbb Q}$ as the set of complex numbers that are roots of a nonzero polynomial with rational ...

16
votes

Accepted

### Non-commutative Galois theory

Let $k$ be a field. Say that a $k$-algebra $A$ is separable if any of the following equivalent conditions holds (it is not obvious that they are equivalent):
$A$ is projective as an $(A, A)$-bimodule....

16
votes

Accepted

### Structure of coefficients of polynomials giving a specified Galois group

Every Galois group is Zariski dense, so that's not a very exciting measurement. In fact, for any degree n field, minimal polynomials of generators of that field are Zariski dense. This follows from ...

15
votes

Accepted

### A Galois extension over $\mathbb{Q}$ with Galois group $A_4$ and with cyclic decomposition groups

Yes. Note that Daniel Loughran's comment to David Speyer's answer to this question states that for any solvable group $G$, there is a Galois extension $L/\mathbb{Q}$ with all decomposition groups ...

15
votes

Accepted

### Splitting of polynomials over rational function fields

Your question 1 is open and is equivalent to the problem of determining the rank of an elliptic curve over a number field. (In most "practical cases" it should be solvable.) In particular, we don't ...

15
votes

Accepted

### Centraliser of an absolute Galois group

For an extension $L/\mathbf{Q}_p$, let $G_L$ denote the absolute Galois group $\mathrm{Gal}(\overline{L}/L)$.
If $\sigma \in G_{\mathbf{Q}_p}$ acts centrally on $G_K$, then it also acts centrally on ...

15
votes

Accepted

### Is $\text{PSL}_2(\mathbb{F}_{p^m})$ known to be a Galois group over $\mathbb{Q}$ for $m>1$?

It is known that ${\rm PSL}_{2}(\mathbb{F})$ can be realized for some $\mathbb{F}$ but definitely not all at present. According to David Zywina's note here, ${\rm PSL}_{2}(\mathbb{F}_{27})$ is the ...

15
votes

Accepted

### Representing finite sums of rational powers of 2

If $A$ and $B$ are distinct finite subsets of $\mathbb Q$ which are not both subsets of $\mathbb Z$, let $d$ be the least common denominator and $m$ the minimum of $A \cup B$. Thus $\sum_{a \in A} 2^...

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