All Questions

In Serre's Corps Locaux, Chapter 2 §3, is presented a classical proof. We are in an "ABKL" setup, where $K/L$ is finite, $A$ is Dedekind, $B$ is the integral closure of $A$ and $B$ is $A$-...

Suppose $ K $ be an algebraic number field and $ n $ be an even integer. Is it possible to find atleast one $ p $ such that $ p\equiv 1( \text{mod}~ n)$ and $ \mathbb{Q}(\eta_p) $ is linearly disjoint ...

Let $K$ be a field. We consider the polynomial $f(x) = x^4 + 1$. It is known that $f(x)$ is irreducible over $\mathbb{Q}$ but reducible over any finite field. Thus $ f(x)$ is reducible over any field $...

In my research it would be great if the following result is valid. In what follows, $\overline{\mathbb{Q}}$, $\overline{\mathbb{Q}}_n$ and $\overline{\mathbb{Q}}_{<n}$ denotes the set of algebraic ...

Actually I am interested in degree $ 8 $ cyclic extension over $ \mathbb{Q} $. Let $ L $ be such extension. At first I was thinking to take basis as normal basis, as we can determine the galois group ...

Let $A$ be a $\mathbf{Q}$-algebra. We say $A$ is "residually abelian", if there exists a maximal ideal $\mathfrak{m}$ of $A$ whose residue field $\kappa(\mathfrak{m})$ is a Kummer extension of an ...