Completion of a valued field
= Completion of a valued field
{title2=$\widehat K$}
The metric completion of a <field> for an <absolute value on a field> is formed from Cauchy sequences modulo sequences tending to zero. Addition and multiplication extend continuously, and every nonzero completed element is invertible, so the result remains a <field>. The <absolute value on a field> extends uniquely. A place of a <number field> determines the corresponding completion $K_v$.