Transformation Into Algebra
All rational numbers are constructible, and all constructible numbers are algebraic numbers. Also, if a and b are constructible numbers with b ≠ 0, then a − b and a/b are constructible. Thus, the set K of all constructible complex numbers forms a field, a subfield of the field of algebraic numbers.
Furthermore, K is closed under square roots and complex conjugation. These facts can be used to characterize the field of constructible numbers, because, in essence, the equations defining lines and circles are no worse than quadratic. The characterization is the following: a complex number is constructible if and only if it lies in a field at the top of a finite tower of quadratic extensions, starting with the rational field Q. More precisely, z is constructible if and only if there exists a tower of fields
where z is in Kn and for all 0 ≤ j < n, the dimension = 2.
Read more about this topic: Constructible Number
Famous quotes containing the word algebra:
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)