Finite Codimension Subfields of a Field Complete with Respect to a Real Valuation

ABSTRACT

Let K be a field complete with respect to a real valuation v and not algebraically closed. We will show that every finite codimension subfield of K is closed in the v-adic topology if and only if the degree of imperfection of K is finite. It follows that there are incomplete finite codimension subfields of K when the degree of imperfection of K is infinite. These examples exhibit other interesting pathologies. We are able to give a necessary (and in the case of a discrete real valuation also sufficient) condition for a given finite codimension subfield to be complete. Finally, we give some applications to fields of Laurent series.

Communicated by A. Prestel.

Keywords:

• Complete fields
• Degree of imperfection
• Laurent series
• Nonarchimedean absolute values
• Real valuations

Mathematics Subject Classification:

• Primary 12J25
• 16W60
• Secondary 12F15
• 12J10
• 12J20
• 13B35
• 13J10
• 16W25

ACKNOWLEDGMENT

We would like to thank the referee for his useful comments and for suggesting Example 14.