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Sat May 30 16:24:10 PDT 1992 REDUCE 3.4.1, 15-Jul-92 ... 1: 1: 2: 2: (ARNUM) 3: 3: Time: 68 ms 4: 4: % 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 W6 := 64*(X + ---*A*X + ---*A )*(X - ---*A*X + ---*A )*(X + ---*A) 2 4 2 4 2 1 *(X - ---*A) 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; 1 3 1 2 1 1 1 3 W7 := 16*(X + (---*A + ---*A + ---*A + ---))*(X - ---*A ) 2 2 2 2 2 1 2 1 *(X - ---*A )*(X - ---*A) 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 - (------*A1 + ------*A1 - ------*A1 - ------*A1 + ------*A1 8243 8243 8243 8243 8243 6825 - ------) 8243 sqrt5; 120 5 27 4 2000 3 1170 2 14919 ------*A1 + ------*A1 - ------*A1 - ------*A1 + -------*A1 8243 8243 8243 8243 8243 6825 - ------ 8243 sqrt(x**2+2*(sqrt5-cbrt3)*x+5-2*sqrt5*cbrt3+cbrt3**2); 240 5 54 4 4000 3 2340 2 21595 X + (------*A1 + ------*A1 - ------*A1 - ------*A1 + -------*A1 8243 8243 8243 8243 8243 13650 - -------) 8243 on rationalize; 1/(x+sqrt5-cbrt3); 5 240 5 54 4 4000 3 2340 2 21595 (X - (------*A1 + ------*A1 - ------*A1 - ------*A1 + -------*A1 8243 8243 8243 8243 8243 13650 4 108 5 800 4 1800 3 - -------)*X - (------*A1 - ------*A1 - ------*A1 8243 8243 8243 8243 15433 2 15900 14465 3 3 2 + -------*A1 + -------*A1 + -------)*X - (A1 - 15*A1)*X - ( 8243 8243 8243 900 5 3919 4 15000 3 8775 2 148986 ------*A1 - ------*A1 - -------*A1 - ------*A1 + --------*A1 8243 8243 8243 8243 8243 154225 1919 5 1050 4 18245 3 - --------)*X - (------*A1 + ------*A1 - -------*A1 8243 8243 8243 8243 12528 2 236725 73080 6 4 3 - -------*A1 + --------*A1 - -------))/(X - 15*X - 6*X 8243 8243 8243 2 + 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 1 3 7 2 11 4 {X - (----*A + ----*A + ---*A - ----*A + ----*A + ---), 12 12 6 12 12 3 1 5 1 4 2 3 1 2 2 7 X + (---*A + ---*A + ---*A - ---*A + ---*A + ---), 6 3 3 6 3 3 1 5 1 4 1 3 5 2 1 X - (----*A + ---*A + ---*A + ----*A - ---*A + 1)} 12 4 2 12 4 end; 5: 5: Time: 17170 ms plus GC time: 595 ms 6: 6: Quitting Sat May 30 16:24:29 PDT 1992