Let be any integer which is not the perfect square of another integer. Let us abbreviate and define

This is called the quadratic extension of the rationals associated to . It is an extension since it contains the field as a subset. It is called an imaginary quadratic field if and a real quadratic field if .

For brevity, let . Multiplication in is given by

for . Inverse is given by

Since is not a perfect square, for rational unless .

Subsections

david joyner 2008-04-20