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Results tagged with number-fields
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user 35416
4
votes
Accepted
Dihedral extension unramified at primes dividing order of group?
$\DeclareMathOperator\Gal{Gal}$The answer is "yes", and this is an easy exercise in class field theory: if, for example, $q$ is a prime number that is $1\pmod n$, and $F$ is a quadratic number field i …
- 13k
8
votes
Galois embedding question for dihedral groups
The answer is "no", in general, since there may be local obstructions. Suppose, for example, that $k$ and $n$ are odd prime powers, and let $L/\mathbb{Q}$ be the unique intermediate quadratic in $F$. …
- 13k
3
votes
Why is every quadratic subfield of a Galois extension of the rationals with the quaternions ...
Here is a solution using some representation theory of the quaternions and Dirichlet's theorem on the units:
the quadratic subfields of our quaternion extension (call it $F$) are of the form $\mathbb{ …
- 13k