Pari-gp 2006/7/ Pari-gp 2. Microsoft Windows Pari-gp Galois 8.

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1 Pari-gp 2006/7/12 1 Pari-GP

2 Pari-gp 2006/7/ Pari-gp 2. Microsoft Windows Pari-gp Galois 8.

3 Pari-gp 2006/7/ ε.

4 Pari-gp 2006/7/12 3 Pari-gp Pari C gp GP gp K. Belabas users manual 200 tutorial 50

5 Pari-gp 2006/7/ tutorial GAP 50

6 Pari-gp 2006/7/12 4 Pari-gp GNU Public License Linux, FreeBSD Unix clone, Apple MacOS X, Microsoft Windows

7 Pari-gp 2006/7/12 5 Pari-gp Q: A:

8 Pari-gp 2006/7/12 6 Pari-gp Q: A: gp 4MB 500,000

9 Pari-gp 2006/7/12 7 Pari-gp Q: C gp A: gp for() C gp2c gp C

10 Pari-gp 2006/7/12 8 Pari-gp Q: A: Abel Smith Normal Form, Abel Hermite Normal Form,

11 Pari-gp 2006/7/12 9 Pari-gp Q: A: H. Cohen GTM 138, GTM 193

12 Pari-gp 2006/7/12 10 Pari-gp Q: A: GRH GRH Heuristic 2

13 Pari-gp 2006/7/12 11 Microsoft Windows 1. Microsoft Windows Pari exe. 2.

14 Pari-gp 2006/7/12 12 gp p =

15 Pari-gp 2006/7/ Pari Perl (Math::Pari), Python, Lisp (CLISP) Risa/Asir

16 Pari-gp 2006/7/12 13 gp exact GMP p

17 Pari-gp 2006/7/ \p \ps. gp? \p 56 56

$13-1 \p$

18 Pari-gp 2006/7/12 14 gp [1,2,3] : [1,2,3]~ : [1,2,3;4,5,6;7,8,9] = *: [1,2,3]*[1,2,3]~ = 14

19 Pari-gp 2006/7/ matdet(), matrank(), mateigen(), charpoly().

20 Pari-gp 2006/7/12 15 gp? a = 1;? a = [1, 2, 3];? a[1] 1

21 Pari-gp 2006/7/12 16 sin(): p gp> sin(i) time = 0 ms. %12 = 0.E *I (e 1 e)/2i = I.

22 Pari-gp 2006/7/12 17 factorint(): MPQS, ECM, ρ, etc... factor(): 1

23 Pari-gp 2006/7/ fermat(n)=2^(2^n)+1. sizedigit(n) n 10 debugging level 3 \g 3

24 Pari-gp 2006/7/12 18 quad qfb quaddisc(x) Q( x) qfbclassno(x) Q( x) Shanks quadclassunit(x) Q( x)

25 Pari-gp 2006/7/12 19 qfbclassno(x)? qfbclassno (-3299) %10 = 27 qfbclassno (-3299,1) %11 = 27 Euler

26 Pari-gp 2006/7/ Qfb(a, b, c) ax 2 + bxy + cy 2.

27 Pari-gp 2006/7/12 20 quadclassunit() quadclassunit(x,{flag}, {tech=[]}) Q( x) Buchmann-McCurley sub-exponential D < 10 25, D > qfbclassno()? quadclassunit(-10^25-3) %20 = [ , [ , 2], [Qfb(7, 1, ), Qfb(13, 13, )], 1]

28 Pari-gp 2006/7/12 21 quadclassunit(x) flag=1 D > 0 tech=[c 1, c 2 ] c 1, c 2 > 0 c 2 = c 0.1 c 2.0 c = 6 GRH GRH bnf...

29 Pari-gp 2006/7/ ? quadclassunit(-3299,,[6,6]) %21 = [27, [9, 3], [Qfb(3, 1, 275), Qfb(23, -17, 39)], 1] GRH

30 Pari-gp 2006/7/12 22 nf bnf big number field/buchman s number field x = 0 2 Q( 3299)? T = x^ ;? bnf = bnfinit(t);? bnf.clgp %69 [27, [9, 3], [[3, 2; 0, 1], [23, 8; 0, 1]]] bnf.no= =27 bnf.cyc= = Z/9Z Z/3Z, bnf.gen=

31 Pari-gp 2006/7/12 23 bnfinit() bnfcertify() : bnfinit() 1? bnf = bnfinit(t);? bnfcertify(bnf) %10 = 1

32 Pari-gp 2006/7/ Hilbert quadhilbert(d): 2 Q( D) Hilbert Q( D)?quadhilbert(-3299) time = 33 ms. %11 = x^27-125*x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x^ *x + 1

33 Pari-gp 2006/7/12 25 monic Z T nfinit(t), bnfinit(t). T θ Q Q(θ) x x x = 0 K? T = x^4 + 24*x^ *x ;? bnf = bnfinit(t);? bnf.sign %68 [0,4]? bnf.clgp %69 [4, [4], [[7, 4, 5, 6; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]]] bnf.no= = 4, bnf.cyc= 4 bnf.gen=

34 Pari-gp 2006/7/12 26 bnf bnf.zk Z HNF idealprimedec() 2? bnfisprincipal(bnf,idealpow(bnf,bnf.gen[1],4)) %190 = [[0]~, [3, -6, 2, -1]~] bnf bnf.gen[1] 4=bnf.no

35 Pari-gp 2006/7/12 27 polcompositum(pol1, pol2, flag=0: pol1 pol2? polcompositum(x^2+23, polsubcyclo(9,3)); %82 = [x^6 + 63*x^4 + 2*x^ *x^2-144*x ]? qfb 23 3=bnfinit(%82[1]); Q( 23) 9 3 Q( 23) Z 3 1st layer.

36 Pari-gp 2006/7/12 28 Q Galois? G17 = galoisinit(polcyclo(17));? galoisisabelian(g17) %203 = [16]? galoispermtopol(g17,g17.gen[1]) %204 = x^7? galoisfixedfield(g17,g17.gen[1]^4,1) %207 = x^4 + x^3-6*x^2 - x + 1? galoisexport(galoisinit(%)) %208 = "Group((1, 3, 2, 4))"

37 Pari-gp 2006/7/ polcyclo(17) 17 Galois 16 σ 7 σ 4 x^4 + x^3-6*x^2 - x + 1 Q Galois GAP Group((1, 3, 2, 4))

38 Pari-gp 2006/7/12 29

39 Pari-gp 2006/7/12 30 if() gp if(,, ) for() for(,, )? for(d=1,100,if(isfundamental(d), h=qfbclassno(d); if(mod(h,3)==mod(0,3),print(d, ", ", h))))

40 Pari-gp 2006/7/12 31 (,...) = { }

41 Pari-gp 2006/7/12 32 file \r file read(file ) write(file,...)... file

42 Pari-gp 2006/7/12 33 ray class group/ray class field bnr rnf gp2c: gp C gp C Unix

43 Pari-gp 2006/7/12 34

2.4.2 if Par

2.4.2 if Par Pari-GP 1 Pari-gp 2 1.1 Pari-gp............... 3 2 gp 4 2.1 gp............................. 5 2.1.1 gp................... 5 2.1.2...................... 6 2.1.3............... 7 2.1.4......................