The Computational Complexity of Division in Quadratic Extension Fields
Thomas Lickteig · SIAM Journal on Computing · 1987
We show that, for any quadratic field extension $k \subset K$, the division in K requires exactly six essential multiplications and divisions over k. In particular, complex division in real arithmetic requires six multiplications and divisions. If additions and subtractions are counted as well, exactly nine operations are required if $k \subset K$ is a radical extension; otherwise ten operations are required.