On the unit groups and the ideal class groups of certain cubic number fields

Key Points

• The research aims to explore the unit group of cubic number fields and identify fundamental units under specific conditions.
• Analyzed the cubic polynomial f(x) = x³ + 3x + a³ with a belonging to integers (Z).
• Investigated the unit group of the field generated by the real root of the polynomial.
• Considered conditions under which specific values become fundamental units.
• Identified that u = 1 - a² - a serves as a fundamental unit under certain conditions.
• Examined the structure of the 3-class group of the cubic number field.