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Results tagged with galois-theory
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user 2481
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.
16
votes
Cyclic extensions
An example is given by any cubic Galois extension of $\mathbb{Q}$, e.g. the extension $\mathbb{Q}(\alpha)$ where $\alpha = \cos \frac{2\pi}{9}$ is a root of $x^3 - 3x + 1$.
This works because if the …
- 37k
4
votes
Accepted
When is $-1$ in the image of a field norm?
Global question: If $p = 1 \bmod 4$, then the element $-1 \in F^\times$ is not a norm from $K^\times$, because $F^\times$ is totally real and $K^\times$ is totally complex; thus $-1$ is not a local no …
- 37k
2
votes
Accepted
Points $\alpha_n$ of $A$ over the $m_n$-th layer in a $\mathbb{Z}_p$-ext. of $K$, where $A$ ...
I suspect that the answer is "no". If you take an elliptic curve with supersingular reduction at $p$, $K$ a sufficiently carefully chosen imaginary quadratic field, and $L / K$ the anticyclotomic $\ma …
- 37k
10
votes
Accepted
Understanding absolute Galois group from its representations
The slogan "number theorists aim to understand $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$" is one that gets used a lot, but it's perhaps a tiny little bit misleading.
Understanding the s …
- 37k