The galois-theory tag has no usage guidance.

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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 ...

**12**

votes

**2**answers

1k views

### 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 ...

**25**

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**4**answers

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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 ...

**5**

votes

**1**answer

490 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 ...

**23**

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**2**answers

540 views

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

This is somewhat similar to this recent question, but extending in a different direction.
Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...

**2**

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**0**answers

182 views

### Parametric Solvable Septics?

Known parametric solvable septics are,
$$x^7+7ax^5+14a^2x^3+7a^3x+b=0\tag{1}$$
$$x^7 + 21x^5 + 35x^3 + 7x + a(7x^6 + 35x^4 + 21x^2 + 1)=0\tag{2}$$
$$x^7 - 2x^6 + x^5 - x^4 - 5x^2 - 6x - 4 + n(x - ...

**27**

votes

**5**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 ...

**42**

votes

**4**answers

4k views

### Degree of sum of algebraic numbers

This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer.
Let $a$ and $b$ be algebraic numbers, with respective degrees ...

**34**

votes

**1**answer

1k 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 ...

**18**

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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 ...

**14**

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**2**answers

1k views

### 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, ...

**15**

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**1**answer

987 views

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

Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$?
The standard irreducibility criteria seem to fail.

**14**

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**5**answers

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### Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$?

Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? I am going to be on the road soon, so pleas don't be offended if I don't respond quickly to a comment.

**13**

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**2**answers

2k views

### 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 ...

**13**

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**2**answers

473 views

### Which polynomials arise as formulas for a conjugate

For any integer $r \geq 2$, et $V_r$ be the set of polynomials $Q \in {\mathbb Q}[X]$ of degree $r-1$ such that there is an algebraic number $\alpha$ of degree $r$ , such that
$Q(\alpha)$ is a ...

**8**

votes

**2**answers

1k views

### 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 ...

**5**

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**4**answers

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### 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 ...

**12**

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**1**answer

1k views

### Is Lehmer's polynomial solvable?

The degree 10 polynomial
$$\displaystyle x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$$
given by D.H. Lehmer in 1933 has the property that its largest real root, $\beta = 1.176280 \cdots$ is ...

**7**

votes

**2**answers

537 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**

votes

**1**answer

688 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 ...

**6**

votes

**3**answers

587 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 ...

**16**

votes

**1**answer

810 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 ...

**11**

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**0**answers

316 views

### Evaluating products of cyclotomic polynomials at roots of unity

Are there general non-trivial conditions on $p(\cdot)$ and $n$, where $p(\cdot)$ is a product of cyclotomic polynomials and $n$ is a positive integer, such that all the coefficients of $p(\cdot)$ are ...

**8**

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**1**answer

649 views

### Galois group of constructible numbers

Let $\mathcal{C}$ be the field of constructible numbers, that is, the complex numbers constructible by compass and straightedge. It can be shown that it consists of all the numbers obtainable by ...

**4**

votes

**1**answer

677 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

**1**answer

830 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 ...

**12**

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**0**answers

292 views

### Aligned roots of irreducible polynomials

It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...

**5**

votes

**2**answers

544 views

### Is there a $\mathbb{Q}$-linear map $T$ over $\mathbb{Q}[x]$ such that for all polynomials $Gal(T(f))$ $\simeq$ The commutator subgroup of $Gal(f)$?

I asked this question at MSE but I did not receive an answer. So I ask it at MO:
We denote the field of rational numbers by $\mathbb{Q}$. The Galois group of a polynomial $f$ is denoted by ...

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votes

**3**answers

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### Solving z^n=a+ib using only radicals of positive real numbers

Let $a+bi\in\mathbf{C}$ be a complex number with $a,b\in\mathbf{R}$. Then
it is easy to find exact solutions of $z^2=a+ib$. For example let $z=u+iv$. Then
$$
u^2+v^2=z\overline{z}=\sqrt{a^2+b^2}
$$
...

**4**

votes

**0**answers

213 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 + ...

**5**

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**0**answers

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### Endofunctors on the category of groups which are Galois- related to a linear map on $\mathbb{Q}[x]$

I thank J.C. and Arturo Magidin for interesting suggestions in the comments to this question
In this post the field of rational numbers is denoted by $\mathbb{Q}$. The space of polynomials with ...

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votes

**2**answers

204 views

### Minimal fields of isomorphism for varieties

Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...

**3**

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**0**answers

557 views

### Differential Galois number theory

Following http://math.stackexchange.com/questions/635659/can-the-error-term-involved-in-the-pnt-be-expressed-in-a-galois-theoretic-framew?noredirect=1#comment1341143_635659, I vainly tried to find ...

**2**

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**2**answers

403 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 $\mathbb{Q}$. ...