4
$\begingroup$

In Engler's valued fields, exercise 3.5.2 goes as follows

Construct valuations on $\Bbb{C}$ of rank $\kappa$ for every cardinal $\kappa \leq 2^{\aleph_0}$.

The idea behind this (for any countably infinite rank) is that there are two kinds of transcendental valuation extensions, the Gauss extension, which does not change the rank of the value group, and the one that increases the rank of the value group by $1$. Also, algebraic extensions do not change the rank of the value group. So since the only extension on $\Bbb{Q}$ is the $p$-adic extension of rank $1$, we can use induction to obtain valuations on $\Bbb{C}$ for any countably infinite rank by taking a transcendence basis for $\Bbb{R}/\Bbb{Q}$.

I'm not very familiar with transfinite induction however. So how does one use transfinite induction to get valuations of uncountably infinite rank?

  • 0
    The rank is just the Krull dimension of the valuation ring.2012-06-22

0 Answers 0