Editing Quadratic field (section)

==Quadratic subfields of cyclotomic fields==

===The quadratic subfield of the prime cyclotomic field===
A classical example of the construction of a quadratic field is to take the unique quadratic field inside the [[cyclotomic field]] generated by a primitive <math>p</math>th root of unity, with <math>p</math> an odd prime number. The uniqueness is a consequence of [[Galois theory]], there being a unique subgroup of [[Index of a subgroup|index]] <math>2</math> in the Galois group over <math>\mathbf{Q}</math>. As explained at [[Gaussian period]], the discriminant of the quadratic field is <math>p</math> for <math>p=4n+1</math> and <math>-p</math> for <math>p=4n+3</math>. This can also be predicted from enough [[Ramification (mathematics)|ramification]] theory. In fact, <math>p</math> is the only prime that ramifies in the cyclotomic field, so <math>p</math> is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants <math>-4p</math> and <math>4p</math> in the respective cases.

===Other cyclotomic fields===
If one takes the other cyclotomic fields, they have Galois groups with extra <math>2</math>-torsion, so contain at least three quadratic fields.  In general a quadratic field of field discriminant <math>D</math> can be obtained as a subfield of a cyclotomic field of <math>D</math>-th roots of unity.  This expresses the fact that the [[Conductor (class field theory)|conductor]] of a quadratic field is the absolute value of its discriminant, a special case of the [[conductor-discriminant formula]].