98
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 ...

- 67.1k

75
votes

### Why is differential Galois theory not widely used?

The theory of differential Galois theory is used, but in algebraic, not differential geometry, under the name of D-modules. A D-module is an object that is somewhat more complicated than a ...

- 119k

68
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

49
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 ...

- 43.8k

45
votes

### Does one real radical root imply they all are?

The answer to the question is yes (so the answer to the title is no) and I will give an example later.
Let me first recall a couple of results. The first one is the following, that can be found in [...

- 62.5k

44
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) ...

- 110k

36
votes

### Why is differential Galois theory not widely used?

As indicated by KConrad in his comments, differential Galois theory is used in the part of transcendental number theory that tries to establish algebraic/linear independence of values of special ...

- 12.5k

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 ...

- 198k

33
votes

Accepted

### What justification can you give for the fact that "most ODEs do not have an explicit solution"?

If the ODE is linear --and the notion of «explicit» refers to Liouvillian solutions (towers of iterated quadrature and exponential of meromorphic functions)-- then its differential Galois group (...

- 5,040

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 ...

- 878

29
votes

Accepted

### Is $x^{n}-x-1$ irreducible?

This is true; it is due to Selmer. Ljunggren (On the irreducibility of certain trinomials and quadrinomials, Math. Scand. 1960) has obtained the complete list of reducible trinomials with $\pm 1$ ...

- 13.5k

29
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, ...

- 391

28
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 ...

- 119k

27
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, ...

- 958

26
votes

Accepted

### Is Lehmer's polynomial solvable?

Lehmer's polynomial is symmetrical,
so $x + x^{-1} =: y$ satisfies a polynomial of half the degree.
It turns out that this is the quintic $y^5 + y^4 - 5y^3 - 5y^2 + 4y + 3 = 0$,
whose Galois group is ...

- 72.6k

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 $\...

- 16.3k

25
votes

Accepted

### Degree 17 number fields ramified only at 2

Thank you for calling this problem to my attention.
I computed $K$ en route to AWS (though this year's
topics
are a rather different flavor of number theory...).
After some simplification (gp's $\rm ...

- 72.6k

24
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 ...

- 937

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 ...

- 67.1k

23
votes

Accepted

### Is there an algebraic number that cannot be expressed using only elementary functions?

I addressed this exact question in my American Mathematical Monthly paper,
What is a closed-form number? Corollary 1 in that paper states that if Schanuel's conjecture holds, then the EL numbers (i.e.,...

- 67.1k

23
votes

Accepted

### What's so special about these $17$th deg equations?

Many of the polynomials in the Klueners-Malle database and also in my
database with John Jones come from families in the way you correctly
describe. So you have "reverse engineered" the source family....

- 246

23
votes

Accepted

### Profinite groups as absolute Galois groups

This is a very good question which is a big open problem. There are a number of theorems, some of them easy and some very difficult, and also a number of conjectures, restricting the class of groups ...

- 14.5k

23
votes

Accepted

### Concrete Applications of knowing $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

Of course, Galois theory intervenes as a basic tool in the study of some diophantine equation. But deeper aspects, in the form of Galois representations, were crucial for the proof of at least 4 ...

- 12.5k

23
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 &...

- 27.6k

22
votes

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

This article from Le Monde (in French) and this blog (in English) are recent and seem to accurately sum up the state of affairs: In April 2012 Jean Malgoire (see the video interview in the blog) ...

- 153k

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 ...

- 7,782

20
votes

Accepted

### Are there irreducible polynomials with all zeros on two concentric circles?

Dubickas and Smyth (On the Remak Height, the Mahler Measure and Conjugate Sets of Algebraic Numbers Lying on Two Circles, 2001) discuss what they call extended Salem numbers.
Moreover, they present ...

- 7,296

19
votes

Accepted

### When complex conjugation lies in the center of a Galois group

Your condition is that $K$ be a kroneckerian field, namely either a totally real (as you mention) field or a totally imaginary quadratic extension of a totally real field: in this second case $K$ is ...

- 5,489

19
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 ...

- 4,621

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