Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On common extensions of valued fields   Given a valuation $v$ on a field $K$, an extension $\bar{v}$ to an algebraic
closure and an extension $w$ to $K(X)$. We want to study the common extensions
of $\bar{v}$ and $w$ to $\bar{K}(X)$. First we give a detailed link between the
minimal pairs notion and the key polynomials notion. Then we prove that in the
case when $w$ is a transcendental extension, then any sequence of key
polynomials admits a maximal element, and in case this sequence does not
contain a limit key polynomial, then any root of the last key polynomial,
describe a common extension.

8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
21
The categories of this paper are: adap-org
28
The categories of this paper are: alg-geom
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con