# Questions tagged [galois-theory]

Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood.

**36**

**2**answers

### Can the unsolvability of quintics be seen in the geometry of the icosahedron?

**5**

**1**answer

### Categorification of covering morphisms

**62**

**9**answers

### Galois Groups vs. Fundamental Groups

**96**

**10**answers

### "Understanding" $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

**75**

**8**answers

### The inverse Galois problem, what is it good for?

**71**

**2**answers

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

**38**

**5**answers

### Computing the Galois group of a polynomial

**31**

**4**answers

### $A_5$-extension of number fields unramified everywhere

**21**

**3**answers

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

**14**

**2**answers

### Frobenius density theorem

**16**

**1**answer

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

**14**

**1**answer

### Non-commutative Galois theory

**15**

**3**answers

### Elementary Luroth theorem proof?

**6**

**4**answers

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

**6**

**2**answers

### Why is $K_{\upsilon}|K$ separable for a global field $K$?

**14**

**2**answers

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

**26**

**2**answers

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

**7**

**1**answer

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

**6**

**1**answer

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

**4**

**1**answer

### Parametric Solvable Septics?

**11**

**4**answers

### What are "nearly initial" objects really called?

**3**

**0**answers

### Differential Galois number theory

**133**

**0**answers

### Grothendieck-Teichmuller conjecture

**61**

**3**answers

### Forcing as a new chapter of Galois Theory?

**58**

**4**answers

### Degree of sum of algebraic numbers

**23**

**5**answers

### Anabelian geometry study materials?

**32**

**3**answers

### Fundamental groups of topoi

**29**

**2**answers

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

**37**

**1**answer

### Degree 17 number fields ramified only at 2

**23**

**5**answers

### Grothendieck's Galois theory without finiteness hypotheses

**26**

**1**answer

### The Galois group of a random polynomial

**14**

**2**answers

### Solvability in differential Galois theory

**26**

**2**answers

### How to solve a quadratic equation in characteristic 2 ?

**14**

**5**answers

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

**19**

**4**answers

### Local inverse Galois problem

**18**

**3**answers

### Profinite groups as absolute Galois groups

**12**

**1**answer

### Abelian $\ell$-adic representations in $\widehat{\mathrm{SL}(2,\mathbb{Z})}$

**9**

**1**answer

### Which of these 4 definitions of Galois coverings of integral schemes are equivalent?

**12**

**2**answers

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

**5**

**4**answers

### Consequences of the Inverse Galois Problem

**24**

**4**answers

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

**16**

**2**answers

### Images of polynomials

**10**

**1**answer

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

**10**

**1**answer

### Unique factorisation of prime geodesics?

**6**

**2**answers

### Frobenius elements in infinite extensions

**15**

**2**answers

### Which polynomials arise as formulas for a conjugate

**8**

**2**answers

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

**6**

**3**answers

### Constructing quintic number fields with certain splitting behaviour

**17**

**1**answer

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

**12**

**1**answer