Basis discrepancies for extensions of valued fields

Let F be a field complete for a real valuation. It is a standard result in valuation theory that a finite extension of F admits a valuation basis if and only if it is without defect. We show that even otherwise, one can construct bases in which the discrepancy between measuring valuation of an element versus on the components in its basis decomposition can be made arbitrarily small. The key step is to verify this for extensions of degree equal to the characteristic by a direct calculation.