A totally real number field is a type of number field, which is defined as a finite extension of the field of rational numbers \( \mathbb{Q} \). Specifically, a number field \( K \) is called totally real if every embedding of \( K \) into the complex numbers \( \mathbb{C} \) maps \( K \) into the real numbers \( \mathbb{R} \).
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A number field is totally real when every embedding into the complex numbers has image in the real numbers. The maximal real subfield of a cyclotomic field is a basic example. Ordinary Hilbert class fields split at real places and therefore retain this property.