All Questions

In his paper "A Hasse principle for function fields over PAC fields" (DOI link), Ido Efrat proves the following result: Let $F$ be an extension of a perfect PAC field $K$ of relative ...

Consider the extension $\mathbb Q(a,b)$ of the field of rationals, where $a$, $b$ are algebraically independent transcendentals. To $\mathbb Q(a,b)$ adjoin the roots of the polynomials $x^5+a^5=1$ and ...

Is there any known example of a perfect pseudo algebraically closed field of positive characteristic containing $\overline{\mathbb{F}_p}$ but is not algebraically closed?

Let $\mathbb{K}$ be an arbitrary field with a subfield $\mathbb{F}$ of index 2. Let $a,b\in\mathbb{K}[X]$ be univariate non-vanishing polynomials over $\mathbb{K}$ of degree $\leq 3$ each. Edit: Due ...

There are quadratic solutions to $x^4+y^4 = z^4$ in $\mathbb{Q} (\sqrt{-7})$. But for equations such as $x^4+y^4 = nz^4$ where $n \in \mathbb{N}, \ n \neq 1$ do there still exist extension fields of $\...

Consider the transcendental extension Q(t) of the field of rationals. To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting field Q(t)[x] is a radical extension of Q(t). Is it true that ...