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 ...
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 [...
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) ...
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 ...
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 (...
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 ...
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 $\pm 1$ ...
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 ...
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
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 ...
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 ...
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....
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 ...
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) ...
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 ...
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
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 ...
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 ...
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 ...

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