Let K/Q be a field, probably not a finite extension. Is it possible for a polynomial to be irreducible over K but have a root in every completion of K? What about all but finitely many completions? ...
Let $L:K$ be a field extension. Let $A$ be a set of elements in $L$, all of which are algebraic over $K$. Construct the field extension $M=K(A)$. I have two questions:  Is $M:K$ an algebraic field ...
This question may be too detailed but perhaps somebody knows the answer: Neukirch proofs in his algebraic number theory book in chapter IV, proposition 6.2, that his class field axiom implies that the ...
Which computer algebra system should I be using to solve large systems of sparse linear equations over a number field?
This is related to Noah's recent question about solving quadratics in a number field, but about an even earlier and easier step. Suppose I have a huge system of linear equations, say ~10^6 equations ...
Suppose I have a bunch of polynomial equations with coefficients in a number field, and suppose further that I'm guaranteed a priori that they have a solution in that number field. Can I leverage ...
Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?" There seems to be a general philosophy that problems over function fields are easier to ...