New answers tagged galois-theory
1
vote
Algebraic numbers which prescribed degree which does not belong to some fields
Proposition 2 is false, although perhaps only for $n = 4$ (and $t = 2$ or $t = 3$). If $t = 2$, every algebraic number $\gamma$ of degree $4$ is contained in $K_{4}$.
Let $K$ be the Galois closure of $...
- 17.7k
5
votes
Algebraic numbers which prescribed degree which does not belong to some fields
Proposition 1 is true.
Let $x$ be an element of degree $n$ whose Galois group is isomorphic to $S_n$. Let $F$ be a finite subset of algebraic elements of degree $<n$. I claim that $x$ is not in the ...
- 51.7k
1
vote
Accepted
Can a general quintic be solved using inverse beta regularized function?
Yes, it seems, it can. Any general quintic can be reduced to the Bring-Jerrard form $x^5+ax+b=0$.
Then Tyma Gaidash found a solution for the Bring-Jerrard form via Inverse Beta Regularized function:
$...
- 8,444
6
votes
Accepted
Completion of infinite degree extension of perfectoid fields is perfectoid?
I'm assuming you mean infinite algebraic extensions, as otherwise there is no standard way of completing them.
Let $K$ be a perfectoid field, let $L$ be an infinite algebraic extension. Then $L$ ...
- 23.7k
0
votes
Galois groups vs. fundamental groups
I found this introduction by David Corwin very accessible and well explained: https://math.berkeley.edu/~dcorwin/files/etale.pdf.
- 3,804
17
votes
Absolute Galois group, number theory and the Axiom of Choice
There would be no consequences, for two reasons:
As Timothy Chow points out, if we define $\overline{\mathbb Q}$ as the set of complex numbers that are roots of a nonzero polynomial with rational ...
- 116k
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 ...
- 65k
Top 50 recent answers are included