Cubic Field

A cubic field, also called a cubic number field, is a number field of extension field degree three over . It can be written as , where is a root of an irreducible polynomial of polynomial degree three.

A cubic field has number field signature or . In the first case, all three field embeddings into are real embeddings, and the cubic field is called totally real. In the second case, there is one real embedding and one complex conjugate pair of imaginary embeddings, and the cubic field is called complex. Equivalently, the polynomial discriminant of is positive in the totally real case and negative in the complex case (Milne 2020).

For example, adjoining a root of gives a totally real cubic field, whereas adjoining a root of gives a complex cubic field. The corresponding polynomial discriminants are 81 and , respectively. A complex cubic field still has a real embedding, so the word "complex" does not mean that it cannot be embedded in .

Norm-Euclideanity imposes a stronger arithmetic condition on the ring of integers. Bagger et al. (2026) determined all 58 complex norm-Euclidean number fields of extension field degree three. Their field discriminants have absolute value at most 876.