Real Quadratic Field

A real quadratic field is a quadratic field with a squarefree integer. The two embeddings of send to and . Both have image in , so is a totally real field. For an imaginary quadratic field, the two embeddings instead send the square root to a pair of nonreal complex conjugates.

The ring of integers is if and otherwise. The number field discriminant is in the first case and in the second (Conrad, Quadratic Integer Rings). For with , the field norm is

Unlike the ring of integers of an imaginary quadratic field, has infinitely many units. Under the embedding with , there is a unique fundamental unit such that every unit is of the form for (Sutherland 2025). The units in correspond to integer solutions of

The positive sign gives the Pell equation. When , the full ring of integers also admits elements of the form with and odd, so solving the equation in can miss the fundamental unit (Conrad, Generalized Pell Equation).

For example, the fundamental unit of is the golden ratio , which has field norm . Its cube is a unit in .