Abstract: According to Hermite there exists only a finite
number of number fields having a given degree, and a given value of
the discriminant, nevertheless this number is not known generally.
The determination of a maximum number of number fields of degree
10 having a given discriminant that contain a subfield of degree 5
having a fixed class number, narrow class number and Galois group
is the purpose of this work. The constructed lists of the first
coincidences of 52 (resp. 50, 40, 48, 22, 6) nonisomorphic number
fields with same discriminant of degree 10 of signature (6,2) (resp.
(4,3), (8,1), (2,4), (0,5), (10,0)) containing a quintic field. For each
field in the lists, we indicate its discriminant, the discriminant of its
subfield, a relative polynomial generating the field over its quintic
field and its relative discriminant, the corresponding polynomial over
Q and its Galois closure are presented with concluding remarks.