15. Number fields

15.1 Ramification

SAGE can access the Jones database of number fields with bounded ramification and degree less than or equal to 6. It must be installed separately (database_jones_numfield).

First load the database:

sage: J = JonesDatabase()            # requires optional database
sage: J                              # requires optional database
      John Jones's table of number fields with bounded ramification 
      and degree <= 6

List the degree and discriminant of all fields in the database that have ramification at most at 2:

sage: [(k.degree(), k.disc()) for k in J.unramified_outside([2])]            # requires optional database
[(1, 1), (2, 8), (2, -4), (2, -8), (4, 2048), (4, -1024), (4, 512), (4, -2048), (4, 256), (4, 2048), (4, 2048)]

List the discriminants of the fields of degree exactly 2 unramified outside 2:

sage: [k.disc() for k in J.unramified_outside([2],2)]            # requires optional database
[8, -4, -8]

List the discriminants of cubic field in the database ramified exactly at 3 and 5:

sage: [k.disc() for k in J.ramified_at([3,5],3)]            # requires optional database
[-6075, -6075, -675, -135]
sage: factor(6075)
3^5 * 5^2
sage: factor(675)
3^3 * 5^2
sage: factor(135)
3^3 * 5

List all fields in the database ramified at 101:

sage: J.ramified_at(101)                     # requires optional database      
      [Number Field in a with defining polynomial 
         x^2 - 101, 
       Number Field in a with defining polynomial 
         x^4 - x^3 + 13*x^2 - 19*x + 361, 
       Number Field in a with defining polynomial 
         x^5 + 2*x^4 + 7*x^3 + 4*x^2 + 11*x - 6, 
       Number Field in a with defining polynomial 
         x^5 + x^4 - 6*x^3 - x^2 + 18*x + 4, 
       Number Field in a with defining polynomial 
         x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17]

15.2 Class numbers

The class_number is a method associated to a QuadraticField object:

sage: K = QuadraticField(29)
sage: K.class_number()
      1
sage: K = QuadraticField(65)
sage: K.class_number()
      2
sage: K = QuadraticField(-11)
sage: K.class_number()
      1
sage: K = QuadraticField(-15)
sage: K.class_number()
      2
sage: K.class_group()
      Multiplicative Abelian Group isomorphic to C2
sage: K = QuadraticField(401)
sage: K.class_group()
      Multiplicative Abelian Group isomorphic to C5
sage: K.class_number()
      5
sage: K.discriminant()
      401
sage: K = QuadraticField(-479)
sage: K.class_group()
      Multiplicative Abelian Group isomorphic to C25
sage: K.class_number()
      25
sage: K.pari_polynomial()
      x^2 + 479
sage: K.degree()
      2
sage: x = PolynomialRing(QQ).gen()
sage: K = NumberField(x^5+10*x+1, 'a')
sage: K
      Number Field in a with defining polynomial x^5 + 10*x + 1
sage: K.degree()
      5
sage: K.pari_polynomial()
      x^5 + 10*x + 1
sage: K.discriminant()
      25603125
sage: K.class_group()
      Trivial Abelian Group
sage: K.class_number()
      1

See also the link for class numbers at http://mathworld.wolfram.com/ClassNumber.html at the Math World site for tables, formulas, and background information.

For cyclotomic fields, try:

sage: CyclotomicField(3)
      Cyclotomic Field of order 3 and degree 2
sage: CyclotomicField(18)
      Cyclotomic Field of order 18 and degree 6
sage: z = CyclotomicField(6).gen(); z
      zeta6
sage: z^3
      -1

For further details, see the documentation strings in the ring/number_field.py file.

15.3 Integral basis

SAGE can compute a list of elements of this number field that are a basis for the full ring of integers of a number field.

sage: x = PolynomialRing(QQ).gen()
sage: K = NumberField(x^5+10*x+1, 'a')
sage: K.integral_basis()
      [1, a, a^2, a^3, a^4]

Next we compute the ring of integers of a cubic field in which 2 is an ``essential discriminant divisor", so the ring of integers is not generated by a single element.

sage: x = PolynomialRing(QQ).gen()
sage: K = NumberField(x^3 + x^2 - 2*x + 8, 'a')
sage: K.integral_basis()
      [1, a, 1/2*a^2 + 1/2*a]

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