attachment:stuff.sage of days14

   1 # Examine factorization of primes:
   2 # Note that 'x' is not explicitly defined
   3 #  (it's a symbolic variable, set by default).
   4 f = x^2-15 
   5 K.<a> = NumberField(f); K 
   6 OK = K.maximal_order(); OK 
   7 A = 5*OK; A 
   8 FF = A.prime_factors(); FF 
   9 P = FF[0] 
  10 P.is_prime() 
  11 P.ramification_index() 
  12 P.residue_class_degree() 
  13 
  14 # Define a function for your own use
  15 def RamifiedPrimes(K):
  16     import sage.rings.number_field.number_field as NF
  17     if not isinstance(K,NF.NumberField_quadratic):
  18         return -1
  19     D = K.discriminant()
  20     FL = D.factor()
  21     PL = [FL[i][0] for i in xrange(len(FL))]
  22     return PL
  23 
  24 # and use it.
  25 RamifiedPrimes(K)
  26 
  27 #if this is in a file, you can load it:
  28 load /tmp/ram.sage
  29 attach /tmp/ram.sage
  30 
  31 # Define a number field with generator alpha,
  32 #  a root of x^3-2 (shows symbolic usage)
  33 K.<alpha> = NumberField(x^3 - 2); K
  34 alpha^3
  35 
  36 # You can deal with polynomials in a more "official" way (we've
  37 #  clobberred Zx by clobbering 'x', so we have to recreate it.
  38 
  39 Zx.<x> = PolynomialRing(ZZ)
  40 f = x^2+2
  41 L.<beta> = NumberField(f); L
  42 beta^2
  43 K.complex_embeddings()   # K is the cubic field above
  44 L.complex_embeddings()
  45 
  46 # Check what happened to 'x'
  47 type(x)
  48 reset('x')
  49 type(x)
  50 
  51 # Clobber 'x' and see what happens:
  52 x = 1
  53 E.<gamma> = NumberField(x^3-2); E
  54 reset('x')
  55 
  56 # Symbolic Use:
  57 a = sqrt(2); a
  58 K.<b> = QQ[a]; K
  59 b
  60 b^2
  61 (b+1)^2
  62 QQ[a/3 + 5]
  63 
  64 # Be aware of notation:
  65 QQ[b]
  66 # vs.
  67 QQ(b)
  68 # What's the difference?
  69 
  70 # Some Galois Group computations
  71 K.<a>=NumberField(x^3-2)
  72 G = K.galois_group(); G
  73 G.order()
  74 
  75 L.<b> = K.galois_closure(); L
  76 G1 = L.galois_group()
  77 G1.order()
  78 G2 = Hom(L, L)
  79 G2
  80 G2.order()
  81 G2.list()
  82 phi = G2.list()[1]
  83 phi
  84 phi(b^2 + 2/3*b)
  85 
  86 # Class Numbers
  87 L.<a> = NumberField(x^2 + 23)
  88 L.class_number()
  89 
  90 # Orders
  91 K.<I> = NumberField(x^2 + 1)
  92 R = K.order(7*I); R
  93 R.basis()
  94 O=K.maximal_order(); O
  95 R.index_in(O)
  96 
  97 # Relative Extensions
  98 K.<sqrt2> = QuadraticField(2)
  99 R.<X> = K[]; R
 100 L.<a> = K.extension(X^3 + sqrt2*X + 5); L
 101 a^3
 102 a^3 + sqrt2*a
 103 L.galois_group()
 104 L.absolute_degree()
 105 
 106 ### Group Theory/Gap Wrapping
 107 PSL1 = PSL(3,4)
 108 PSL1.order()
 109 PSL1.is_simple()
 110 P2 = PSL1.sylow_subgroup(2)
 111 P2.center().order()
 112 P7 = PSL1.sylow_subgroup(7)
 113 # Now, Suppose we want the cosets of P2 in PSL1.
 114 # Check with the TAB trick...
 115 
 116 # No such luck...
 117 # But we can wrap this in Gap clothing and work with
 118 #  Gap directly
 119 # We wrap up the pieces:
 120 GPSL1 = gap(PSL1)
 121 GP7 = gap(P7)
 122 RC = GPSL1.RightCosets(P7)
 123 len(RC)
 124 C = RC[1]; C
 125 # Note that this is *not* a python list:
 126 C[1]
 127 # It *is* a Gap list:
 128 C.Elements()[2]
 129 type(C)