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Results tagged with galois-theory
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user 5101
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.
11
votes
Which groups are Galois over some p-adic field?
I'll upgrade my comment to an answer.
Any finite Galois extension of $\mathbb{Q}_l$ of degree coprime to $l$ is tamely ramified. In particular, its Galois group is an extension of two cyclic groups. …
- 21.4k
11
votes
1
answer
790
views
Cyclic cubic extensions and Kummer theory
The Galois cohomology group $H^1(\mathbb{Q}, \mathbb{Z}/3\mathbb{Z})$ classifies cyclic cubic extensions $K/\mathbb{Q}$ (specifically: the non-trivial elements correspond to Galois cubic field extensi …
- 21.4k
7
votes
3
answers
845
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 exist …
- 21.4k
6
votes
Accepted
Minimal fields of isomorphism for varieties
Yes if $K=\mathbb{R}$ for example, but no in general.
Namely this fails for curves of genus $1$, over $\mathbb{Q}$, say. Given an elliptic curve $E$ over $\mathbb{Q}$ and a positive integer $d$, a ge …
- 21.4k
4
votes
Applications of the Galois embedding problem
Shafarevich made heavy use of embedding problems in his resolution of the inverse Galois problem for solvable groups. The (naive) viewpoint is that solving the relevant embedding problems allowed him …
- 21.4k