Victor Alexandru, Sever Angel Popescu: On a problem in valuation theory, 107-115

Abstract:

Let $(K,v)$ be a nontrivial Krull valued number field and let $\overline {K}=\overline{\mathbf{Q}}$ be a fixed algebraic closure of $K.$ We say that an extension $L/K$, $L\subset\overline{K},$ is a $v$-extension of $(K,v)$ if $v$ does not split in $L.$

In the 2000s dr. doc. Nicolae Popescu (Institute of Mathematics of the Romanian Academy) stated the following hypothesis:

"An algebraic number field $K$ with a nontrivial Krull valuation $v$ on it cannot have a normal $v$-maximal extension." Trying to solve this problem we considered a Krull valued number field $(K,v)$ with some additional properties and we constructed a class of $v$-maximal extensions of $K$ which are not normal extensions of $K.$ Thus, in general, the above hypothesis is still open.

We also give an example of a nontrivial valued field $(T,w)$ which has a normal $w$-maximal extension. Some other auxiliary results are given on these mysterious mathematical objects, $v$-maximal extensions.

Key Words: Valued fields, Galois groups, Henselian fields, decomposition groups, number fields, $v$-maximal extensions, normality.

2020 Mathematics Subject Classification: Primary 12J10, 12J25, 12F10; Secondary 13A18, 12F99.

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