The galois-theory tag has no wiki summary.

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### Frobenius density theorem

As mentioned by @MichaelZieve in his comment re Quadratic residue, Chebotarev's density theorem was preceded by an allegedly much easier theorem of Frobenius (Mike Zieve is certainly not the only one ...

**44**

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**9**answers

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### Galois Groups vs. Fundamental Groups

In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of ...

**25**

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**4**answers

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### Computing the Galois group of a polynomial

Does there exist an algorithm which computes the Galois group of a polynomial
$p(x) \in \mathbb{Z}[x]$? Feel free to interpret this question in any reasonable manner. For example, if the degree of ...

**25**

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**0**answers

496 views

### Degree 17 number fields ramified only at 2

The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...

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**1**answer

1k views

### The Galois group of a random polynomial

Intuitively, the Galois group (of a splitting field over $\mathbb Q$ of) a polynomial $f\in\mathbb Q[X]$ taken at random is most probably the full permutation group on the roots of $f$. This intuition ...

**12**

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**2**answers

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### Using higher-order Bring radicals to solve arbitrary polynomials

It is well known that there is no general formula for the solution of the quintic. Of course, what this really means is that there is no general formula that only involves addition, subtraction, ...

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**2**answers

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### Automorphisms of $\mathbb{C}$

Is it true that $G_{\mathbb{Q}}$, the absolute Galois group of $\mathbb{Q}$, is a subgroup of $Aut(\mathbb{C})$ ?
Or a simpler question: can any automorphism of $\overline{\mathbb{Q}}$ be extended to ...

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**2**answers

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### When is sin(r \pi) expressible in radicals for r rational?

Perhaps this question will not be considered appropriate for MO - so be it. But hear me out before you dismiss it as completely elementary.
As the question suggests, I would like to know when ...

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**4**answers

1k views

### method of finding roots of polynominal equations with arithmetic operations and roots and other functions

Lets recall Platonic construction in plane geometry. It is impossible to square a circle using only ruler and callipers. But is also known that it is possible to do it with ruler which has a mark on ...

**7**

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**2**answers

438 views

### Is there an algorithm to compute efficiently the dessin d'enfant from a Belyi pair?

Let $(X,f)$ be a Belyi pair, i.e. a Riemann surface $X$ together with a morphism $f: X \to \mathbb{P}^1$, ramified only in $0,1, \infty$. Grothendieck's dessin d'enfant is the pre-image $G$ of the ...

**7**

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**1**answer

617 views

### Can roots of any polynomial be expressed using Eulerian function?

I encountered an interesting function which is called "Eulerian" by the Wolfram's MathWorld:
$$\phi(q)=\prod_{k=1}^{\infty} (1-q^{k})$$
It is interesting because it is claimed that roots of any ...

**16**

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**1**answer

711 views

### On the solvable octic $x^8-x^7+29x^2+29=0$

The irreducible but solvable octic,
$$x^8-x^7+29x^2+29=0\tag{1}$$
was first mentioned by Igor Schein in this 1999 sci.math post. This does not factor over a quadratic or quartic extension, but over ...

**6**

votes

**3**answers

519 views

### Constructing quintic number fields with certain splitting behaviour

I am looking for number fields $K$ which satisfy the following properties:
$[K:\mathbb{Q}]=5$.
The Galois closure of $K$ has Galois group $S_5$.
For each prime $p$ which ramifies in $K$, there ...

**3**

votes

**1**answer

782 views

### Galois connections

I've been experiencing minor qualms about my preprint "A Galois Connection in the Social Network" (accepted by Mathematics Magazine, pending revisions), and one of them involves the way I describe the ...

**3**

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**1**answer

409 views

### What is the size of the smallest rigid extension field of the complex numbers?

Suppose we consider a rigid extension field $F$, i.e., $\text{Aut}(F) = 1$ over the complex numbers $\mathbb C$. What is the minimal cardinality of $F$? In particular it should hold that in this case ...

**4**

votes

**1**answer

489 views

### Symplectic groups Sp_{2m}(2) as 2-transitive permutation (i.e. Galois) groups

Hello,
I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms.
Consider the block matrices
...

**3**

votes

**0**answers

169 views

### Solvable parametric $7$th and $13$th degree equations using $\eta(q)/\eta(q^p)$?

Q: Why is that some polynomial relations between eta quotients have a solvable Galois group, even if the deg is $n>4$?
For example, we have the well-known modular equation,
$$u^6 - v^6 + ...

**2**

votes

**2**answers

349 views

### Place stabilizers for the absolute Galois Group

Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. ...

**1**

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**0**answers

112 views

### Parametric Solvable Septics?

Known parametric solvable septics are,
$$x^7+7ax^5+14a^2x^3+7a^3x+b=0\tag{1}$$
$$x^7 - 2x^6 + x^5 - x^4 - 5x^2 - 6x - 4 + n(x - 1)x^2(x + 1)^2=0\tag{2}$$
$$x^7 + 7x^6 - 28(n^2 + 27)x^2 + 112(n^2 + ...