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If one takes the other cyclotomic fields, they have Galois groups with extra -torsion, so contain at least three quadratic fields. In general a quadratic field of field discriminant can be obtained as a subfield of a cyclotomic field of -th roots of unity. This expresses the fact that the conductor of a quadratic field is the absolute value of ...
Karl Rubin found a more elementary proof of the Mazur–Wiles theorem by using Thaine's method and Kolyvagin's Euler systems, described in Lang (1990) and Washington (1997), and later proved other generalizations of the main conjecture for imaginary quadratic fields. [2]
For given low class number (such as 1, 2, and 3), Gauss gives lists of imaginary quadratic fields with the given class number and believes them to be complete. Infinitely many real quadratic fields with class number one Gauss conjectures that there are infinitely many real quadratic fields with class number one.
In number theory, the Heegner theorem [1] establishes the complete list of the quadratic imaginary number fields whose rings of integers are principal ideal domains. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number.
In algebraic number theory, a number field is called totally imaginary (or totally complex) if it cannot be embedded in the real numbers. Specific examples include imaginary quadratic fields, cyclotomic fields, and, more generally, CM fields. Any number field that is Galois over the rationals must be either totally real or totally imaginary.
Dirichlet's unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units . [ 1 ]
The simplest, and motivating, example of a CM-field is an imaginary quadratic field, for which the totally real subfield is just the field of rationals. One of the most important examples of a CM-field is the cyclotomic field Q ( ζ n ) {\displaystyle \mathbb {Q} (\zeta _{n})} , which is generated by a primitive nth root of unity .
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields.It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa () (岩澤 健吉), as part of the theory of cyclotomic fields.