was@modular:~\$ me Magma V2.11-10 Tue Feb 22 2005 10:16:21 on modular [Seed = 2768087591] Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/emacs.m" Loading "/home/was/magma/local/init.m" > > R := PolynomialRing(Rationals()); > K := 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); [ ] > Factorization(J); [ ] > J*I; Principal Ideal of OK Generator: [-7, 14] > I*J; Principal Ideal of OK Generator: [-7, 14] > Factorization(11*OK); [ , ] > 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 := 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; [ ] > F := Factorization(3*OK); > F; [ , ] > F := Factorization(5*OK); > F; [ ] > 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 := 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); [ , ] > 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 [0;33m? [0;34m [0m[0;33m? [0;34m [0m[0;33m? [0;34m [0m[0;33m? [0;34m [0m[0mGoodbye! was@modular:~\$