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Results tagged with galois-theory
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user 35416
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.
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
Accepted
Local factors determine Weil representations - proof of the cyclic case
Why is the map ${\rm Gal}(FL/L)\to {\rm Gal}(F/K)$ injective?
Elements of ${\rm Gal}(FL/L)$ are automorphisms of $FL$ that act trivially on $L$. To be in the kernel of the above map means to also act …
- 13k
3
votes
Isomorphism related to the first cohomology group
Each of the quotients in your expression is 0, so yes, there is an isomorphism, but it's between trivial groups. Here is the proof: for a Galois extension $K/\mathbb{Q}$, write down the short exact se …
- 13k
7
votes
What evidence is there that $\mathbb{Q}^{ab}$ is ample?
It seem like most of the evidence is rather indirect. The ampleness of $\mathbb{Q}^{ab}$ would settle other open conjectures/questions:
By a result of Pop, it would imply the following conjecture of …
- 13k
3
votes
Accepted
Proof of a Simple Converse in Algebraic Number Theory
If you throw in the residue degrees as well, then you get Zev's question. Otherwise, the converse is not true. As an example, consider a quadratic field that has an unramified everywhere $A_5$ Galois …
- 13k
6
votes
Accepted
Are all solvable groups *regularly* realizable over Q(x)?
It is known that all finite abelian groups are regularly realisable over $\mathbb{Q}(x)$. See e.g. B.H. Matzat, Konstruktive Galoistheorie, p. 224, M. Fried and M. Jarden, Field Arithmetic, Lemma 24.4 …
- 13k
5
votes
What (permutation) groups can occur as galois groups of irreducible polynomials of degree n
This question is ambiguous and can be understood in various ways, since no mention is made of the base field. If you are asking "for what subgroups $G$ of $S_n$ do there exist fields $L,K$ such that $ …
- 13k
4
votes
Is it possible to recover the degree of a field extension from a list of elements and the gr...
To put this one to rest, I will answer the more precise question that, after much prodding, we got Adam to formulate in the comments. I am merely paraphrasing a comment of Qiaochu.
If you are given t …
- 13k
1
vote
Congruences mod primes in Galois extensions
Edit: sometimes you do get the congruence you want for traces. See corrected answer:
to say that $a\equiv b \; ({\rm mod}\;{\mathfrak P})$ is equivalent to saying $a-b \in {\mathfrak P}$, so that ${\ …
- 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