was@modular:~$ me
Magma V2.11-10    Tue Feb 22 2005 10:16:21 on modular  [Seed = 2768087591]
Type ? for help.  Type <Ctrl>-D to quit.

Loading startup file "/home/was/magma/local/emacs.m"

Loading "/home/was/magma/local/init.m"
> > R<x> := PolynomialRing(Rationals());
> K<a> := NumberField(x^2-5);
> K;
Number Field with defining polynomial x^2 - 5 over the Rational
Field
> OK := RingOfIntegers(K);
> OK;

>> OK;
   ^
User error: Identifier 'OK' has not been declared or assigned
> > OK := RingOfIntegers(K);
> OK;
Maximal Order of Equation Order with defining polynomial x^2 - 5 
over ZZ
> I := 7*OK;
>I;
Principal Ideal of OK
Generator:
    [7, 0]
> OK.1;
OK.1
> OK.2;
OK.2
> K!OK.1;
1
> K!OK.2;
1/2*(a + 1)
> I;
Principal Ideal of OK
Generator:
    [7, 0]
> I^2;
Principal Ideal of OK
Generator:
    [49, 0]
> J := a*OK;
> J;
Principal Ideal of OK
Generator:
    [-1, 2]
> I*J;
Principal Ideal of OK
Generator:
    [-7, 14]
> I+J;
Principal Ideal of OK
Generator:
    [1, 0]
> Factorization(I);
[
    <Principal Prime Ideal of OK
    Generator:
        [7, 0], 1>
]
> Factorization(J);
[
    <Prime Ideal of OK
    Two element generators:
        [5, 0]
        [-1, 2], 1>
]
> J*I;
Principal Ideal of OK
Generator:
    [-7, 14]
> I*J;
Principal Ideal of OK
Generator:
    [-7, 14]
> Factorization(11*OK);
[
    <Prime Ideal of OK
    Two element generators:
        [11, 0]
        [3, 2], 1>,
    <Prime Ideal of OK
    Two element generators:
        [11, 0]
        [6, 2], 1>
]
> I/J;
Fractional Principal Ideal of OK
Generator:
    -7/5*OK.1 + 14/5*OK.2
> (11*OK)*I/(J^2);
Fractional Principal Ideal of OK
Generator:
    77/5*OK.1
> K<x> := NumberField(x^3+4*x^2-5*x+3);
> K;
Number Field with defining polynomial x^3 + 4*x^2 - 5*x + 3 over 
the Rational Field
> GaloisGroup(K);
Permutation group acting on a set of cardinality 3
    (1, 3)
    (1, 2)
[ 8, 3, 2 ]
17
> #$1;
6
> OK := RingOfIntegers(K);
> F := Factorisation(2*OK);
> F;
[
    <Principal Prime Ideal of OK
    Generator:
        [2, 0, 0], 1>
]
> F := Factorization(3*OK);
> F;
[
    <Prime Ideal of OK
    Two element generators:
        [3, 0, 0]
        [0, 1, 0], 1>,
    <Prime Ideal of OK
    Two element generators:
        [3, 0, 0]
        [2, 1, 0], 2>
]
> F := Factorization(5*OK);
> F;
[
    <Principal Prime Ideal of OK
    Generator:
        [5, 0, 0], 1>
]
> Discriminant(K);   // WRONG!
-1191
> Discriminant(RingOfIntegers(K));
-1191
> K;
Number Field with defining polynomial x^3 + 4*x^2 - 5*x + 3 over 
the Rational Field
> MinimalPolynomial(2*K.1);
x^3 + 8*x^2 - 20*x + 24
> L := NumberField(x^3 + 8*x^2 - 20*x + 24);
> R<x> := PolynomialRing(RationalField());
> L := NumberField(x^3 + 8*x^2 - 20*x + 24);
> L;
Number Field with defining polynomial x^3 + 8*x^2 - 20*x + 24 
over the Rational Field
> Discriminant(L);
-76224
> Discriminant(RingOfIntegers(L));
-1191
> Factorisation(76224);
[ <2, 6>, <3, 1>, <397, 1> ]
> Factorisation(1191);
[ <3, 1>, <397, 1> ]
> Factorization(397*OK);
[
    <Prime Ideal of OK
    Two element generators:
        [397, 0, 0]
        [172, 1, 0], 1>,
    <Prime Ideal of OK
    Two element generators:
        [397, 0, 0]
        [313, 1, 0], 2>
]
> quit
> ;

Total time: 0.370 seconds, Total memory usage: 3.46MB
You have mail in /var/mail/was
was@modular:~$ gp
Reading GPRC: /etc/gprc ...Done.

                    GP/PARI CALCULATOR Version 2.2.9 (alpha)
               amd64 running linux (x86-64 kernel) 64-bit version
              compiled: Feb 16 2005, gcc-3.3.5 (Debian 1:3.3.5-8)
                (readline v4.3 enabled, extended help available)

                       Copyright (C) 2003 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and 
comes WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

parisize = 8000000, primelimit = 500000
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Goodbye!
was@modular:~$