Sun Aug 18 16:14:07 2002 run on Windows
% Test of algebraic number package.
defpoly sqrt2**2-2;
1/(sqrt2+1);
sqrt2 - 1
(x**2+2*sqrt2*x+2)/(x+sqrt2);
x + sqrt2
on gcd;
(x**3+(sqrt2-2)*x**2-(2*sqrt2+3)*x-3*sqrt2)/(x**2-2);
2
x - 2*x - 3
--------------
x - sqrt2
off gcd;
sqrt(x**2-2*sqrt2*x*y+2*y**2);
x - sqrt2*y
off arnum;
%to start a new algebraic extension.
defpoly cbrt5**3-5;
on rationalize;
1/(x-cbrt5);
2 2
x + cbrt5*x + cbrt5
-----------------------
3
x - 5
off rationalize;
off arnum;
%to start a new algebraic extension.
%The following examples are taken from P.S. Wang Math. Comp. 30,
% 134,(1976),p.324.
on factor;
defpoly i**2+1=0;
w0 := x**2+1;
w0 := (x + i)*(x - i)
w1 := x**4-1;
w1 := (x + i)*(x - i)*(x + 1)*(x - 1)
w2 := x**4+(i+2)*x**3+(2*i+5)*x**2+(2*i+6)*x+6;
2
w2 := (x + i*x + 3)*(x + i + 1)*(x - (i - 1))
w3 := (2*i+3)*x**4+(3*i-2)*x**3-2*(i+1)*x**2+i*x-1;
2 2 2 3
w3 := (2*i + 3)*(x + i*x - 1)*(x - (----*i - ----))
13 13
off arnum;
defpoly a**2-5;
w4 := x**2+x-1;
1 1 1 1
w4 := (x + ---*a + ---)*(x - (---*a - ---))
2 2 2 2
off arnum;
defpoly a**2+a+2;
w5 := x**4+3*x**2+4;
w5 := (x + a + 1)*(x + a)*(x - (a + 1))*(x - a)
off arnum;
defpoly a**3+2=0;
w6:=64*x**6-4;
2 1 1 2 2 1 1 2 1 1
w6 := 64*(x + ---*a*x + ---*a )*(x - ---*a*x + ---*a )*(x + ---*a)*(x - ---*a)
2 4 2 4 2 2
off arnum;
defpoly a**4+a**3+a**2+a+1=0;
w7:=16*x**4+8*x**3+4*x**2+2*x+1;
w7 :=
1 3 1 2 1 1 1 3 1 2 1
16*(x + ---*a + ---*a + ---*a + ---)*(x - ---*a )*(x - ---*a )*(x - ---*a)
2 2 2 2 2 2 2
off arnum, factor;
defpoly sqrt5**2-5,cbrt3**3-3;
*** Defining polynomial for primitive element:
6 4 3 2
a1 - 15*a1 - 6*a1 + 75*a1 - 90*a1 - 116
cbrt3**3;
3
sqrt5**2;
5
cbrt3;
120 5 27 4 2000 3 1170 2 6676 6825
- (------*a1 + ------*a1 - ------*a1 - ------*a1 + ------*a1 - ------)
8243 8243 8243 8243 8243 8243
sqrt5;
120 5 27 4 2000 3 1170 2 14919 6825
------*a1 + ------*a1 - ------*a1 - ------*a1 + -------*a1 - ------
8243 8243 8243 8243 8243 8243
sqrt(x**2+2*(sqrt5-cbrt3)*x+5-2*sqrt5*cbrt3+cbrt3**2);
240 5 54 4 4000 3 2340 2 21595 13650
x + ------*a1 + ------*a1 - ------*a1 - ------*a1 + -------*a1 - -------
8243 8243 8243 8243 8243 8243
on rationalize;
1/(x+sqrt5-cbrt3);
5 240 5 54 4 4000 3 2340 2 21595 13650
(x - (------*a1 + ------*a1 - ------*a1 - ------*a1 + -------*a1 - -------)
8243 8243 8243 8243 8243 8243
4
*x -
108 5 800 4 1800 3 15433 2 15900 14465 3
(------*a1 - ------*a1 - ------*a1 + -------*a1 + -------*a1 + -------)*x
8243 8243 8243 8243 8243 8243
3 2
- (a1 - 15*a1)*x -
900 5 3919 4 15000 3 8775 2 148986 154225
(------*a1 - ------*a1 - -------*a1 - ------*a1 + --------*a1 - --------)*x
8243 8243 8243 8243 8243 8243
1919 5 1050 4 18245 3 12528 2 236725 73080
- (------*a1 + ------*a1 - -------*a1 - -------*a1 + --------*a1 - -------
8243 8243 8243 8243 8243 8243
6 4 3 2
))/(x - 15*x - 6*x + 75*x - 90*x - 116)
off arnum, rationalize;
split_field(x**3+2);
*** Splitting field is generated by:
6
a3 + 108
1 4 1
{----*a3 + ---*a3,
36 2
1 4
- ----*a3 ,
18
1 4 1
----*a3 - ---*a3}
36 2
for each j in ws product (x-j);
3
x + 2
split_field(x**3+4*x**2+x-1);
*** Splitting field is generated by:
3 2
a4 + 4*a4 + a4 - 1
2 2
{a4,a4 + 3*a4 - 2, - (a4 + 4*a4 + 2)}
for each j in ws product (x-j);
3 2
x + 4*x + x - 1
split_field(x**3-3*x+7);
*** Splitting field is generated by:
6 4 2
a6 - 18*a6 + 81*a6 + 1215
1 4 5 2 1 2
{-----*a6 - ----*a6 + ---*a6 + ---,
126 42 2 7
1 4 5 2 4
- (----*a6 - ----*a6 + ---),
63 21 7
1 4 5 2 1 2
-----*a6 - ----*a6 - ---*a6 + ---}
126 42 2 7
for each j in ws product (x-j);
3
x - 3*x + 7
split_field(x**3+4*x**2+x-1);
*** Splitting field is generated by:
3 2
a7 + 4*a7 + a7 - 1
2 2
{a7,a7 + 3*a7 - 2, - (a7 + 4*a7 + 2)}
for each j in ws product (x-j);
3 2
x + 4*x + x - 1
split_field(x**3-x**2-x-1);
*** Splitting field is generated by:
6 5 4 3 2
a9 - 6*a9 + 7*a9 + 12*a9 - 17*a9 - 6*a9 + 53
3 4 3 3 1 2 5 17
{ - (----*a9 - ----*a9 - ----*a9 - ----*a9 + ----),
76 19 38 38 76
3 4 6 3 1 2 14 17
----*a9 - ----*a9 - ----*a9 + ----*a9 + ----,
38 19 19 19 38
3 4 3 3 1 2 33 59
- (----*a9 - ----*a9 - ----*a9 + ----*a9 - ----)}
76 19 38 38 76
for each j in ws product (x-j);
3 2
x - x - x - 1
% A longer example.
off arnum;
defpoly a**6+3*a**5+6*a**4+a**3-3*a**2+12*a+16;
factorize(x**3-3);
1 5 1 4 2 3 1 2 2 7
{{x + ---*a + ---*a + ---*a - ---*a + ---*a + ---,1},
6 3 3 6 3 3
1 5 1 4 1 3 7 2 11 4
{x - (----*a + ----*a + ---*a - ----*a + ----*a + ---),1},
12 12 6 12 12 3
1 5 1 4 1 3 5 2 1
{x - (----*a + ---*a + ---*a + ----*a - ---*a + 1),1}}
12 4 2 12 4
end;
Time for test: 2955 ms, plus GC time: 140 ms