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nfbasis(x^2+1)
[1, x]
nfbasis(x^4 -14*x^2 + 53)
[1, x, 1/2*x^2 + 1/2, 1/4*x^3 + 1/4*x^2 - 1/4*x - 1/4]
nfdisc(x^2+1)
-4
nfdisc(x^4 -14*x^2 + 53)
3392
Mod(3392,4)
Mod(0, 4)
nfdisc(x^4-90*x^2+53)
206105552
nfbasis(x^4-90*x^2+53)
[1, x, 1/4*x^2 + 1/4, 1/4*x^3 + 1/4*x]
nfbasis(x^4-4*x^2+3)
[1, x, 1/2*x^2 + 1/2, 1/4*x^3 + 1/4*x^2 + 1/4*x + 1/4]
nfdisc(x^4-4*x^2+3)
12
nfbasis(x^4-6*x^2+3)
[1, x, x^2, x^3]
nfdisc(x^4-6*x^2+3)
27648
nfdisc(x^2-1)
1
D=[1,sqrt(7+2*I),I,I*(sqrt(7+2*I));1,sqrt(7-2*I),-1*I,(-1*I)*sqrt(7-2*I); 1,-1*sqrt(7+2*I),I,I*(-1*sqrt(7+2*I));1,-1*sqrt(7-2*I),-1*I,(-1*I)*(-1*sqrt(7-2*I))]
[1 2.6720881244151097674232871083920323733 + 0.37423915433885206387815653913250097460*I I -0.37423915433885206387815653913250097460 + 2.6720881244151097674232871083920323733*I] [1 2.6720881244151097674232871083920323733 - 0.37423915433885206387815653913250097460*I -I -0.37423915433885206387815653913250097460 - 2.6720881244151097674232871083920323733*I] [1 -2.6720881244151097674232871083920323733 - 0.37423915433885206387815653913250097460*I I 0.37423915433885206387815653913250097460 - 2.6720881244151097674232871083920323733*I] [1 -2.6720881244151097674232871083920323733 + 0.37423915433885206387815653913250097460*I -I 0.37423915433885206387815653913250097460 + 2.6720881244151097674232871083920323733*I]
(matdet(D))^2
13568.000000000000000000000000000000000 + 0.E-34*I
nfbasis(x^4-4*x^2+3)
[1, x, 1/2*x^2 + 1/2, 1/4*x^3 + 1/4*x^2 + 1/4*x + 1/4]
nfbasis(x^4-14*x^2+53)
[1, x, 1/2*x^2 + 1/2, 1/4*x^3 + 1/4*x^2 - 1/4*x - 1/4]
nfbasis(x^4-6*x^2+3)
[1, x, x^2, x^3]
nfbasis(x^4-6*x^2+25)
[1, x, 1/8*x^2 + 1/2*x - 3/8, 1/40*x^3 - 11/40*x + 1/2]
nfbasis(x^4-6*x^2+34)
[1, x, 1/5*x^2 + 2/5, 1/5*x^3 + 2/5*x]
nfbasis(x^4-16*x^2+68)
[1, x, 1/4*x^2 + 1/2*x + 1/2, 1/8*x^3 - 1/4*x + 1/2]
nfbasis(x^4-32*x^2+300)
[1, x, 1/4*x^2 + 1/2, 1/20*x^3 - 1/10*x]
nfbasis(x^4-4*x^2+8)
[1, x, 1/2*x^2, 1/4*x^3]
A=[1,sqrt(2+2*I),1+I,1/2*(1+I)*(sqrt(2+2*I));1,sqrt(2-2*I),1-I,1/2*(1-I)*(sqrt(2-2*I));1,-1*sqrt(2+2*I),1+I,1/2*(1+I)*(-1*sqrt(2+2*I));1,-1*sqrt(2-2*I),1-I,1/2*(1-I)*(-1*sqrt(2-2*I))]
[1 1.5537739740300373073441589530631469482 + 0.64359425290558262473544343741820980892*I 1 + I 0.45508986056222734130435775782246856962 + 1.0986841134678099660398011952406783786*I] [1 1.5537739740300373073441589530631469482 - 0.64359425290558262473544343741820980892*I 1 - I 0.45508986056222734130435775782246856962 - 1.0986841134678099660398011952406783785*I] [1 -1.5537739740300373073441589530631469482 - 0.64359425290558262473544343741820980892*I 1 + I -0.45508986056222734130435775782246856962 - 1.0986841134678099660398011952406783786*I] [1 -1.5537739740300373073441589530631469482 + 0.64359425290558262473544343741820980892*I 1 - I -0.45508986056222734130435775782246856962 + 1.0986841134678099660398011952406783785*I]
(matdet(A))^2
512.00000000000000000000000000000000000 + 4.255744136770140078 E-36*I
nfdisc(x^4-4*x^2+8)
512
B=[1,sqrt(2+2*I)/2,I,I*(sqrt(2+2*I));1,sqrt(2-2*I)/2,-I,-I*(sqrt(2-2*I));1,-1*sqrt(2+2*I)/2,I,I*(-1*sqrt(2+2*I));1,-1*sqrt(2-2*I)/2,-I,-I*(-1*sqrt(2-2*I))]
[1 0.77688698701501865367207947653157347408 + 0.32179712645279131236772171870910490446*I I -0.64359425290558262473544343741820980892 + 1.5537739740300373073441589530631469482*I] [1 0.77688698701501865367207947653157347408 - 0.32179712645279131236772171870910490446*I -I -0.64359425290558262473544343741820980892 - 1.5537739740300373073441589530631469482*I] [1 -0.77688698701501865367207947653157347408 - 0.32179712645279131236772171870910490446*I I 0.64359425290558262473544343741820980892 - 1.5537739740300373073441589530631469482*I] [1 -0.77688698701501865367207947653157347408 + 0.32179712645279131236772171870910490446*I -I 0.64359425290558262473544343741820980892 + 1.5537739740300373073441589530631469482*I]
(matdet(B))^2
512.00000000000000000000000000000000000 - 4.255744136770140078 E-36*I
C=[1,sqrt(7+2*I),I,-I*(1+sqrt(7+2*I));1,sqrt(7-2*I),-I,I*(1+sqrt(7-2*I));1,-1*sqrt(7+2*I),I,-I*(1-sqrt(7+2*I));1,-1*sqrt(7-2*I),-I,I*(1-sqrt(7-2*I))]
[1 2.6720881244151097674232871083920323733 + 0.37423915433885206387815653913250097460*I I 0.37423915433885206387815653913250097460 - 3.6720881244151097674232871083920323733*I] [1 2.6720881244151097674232871083920323733 - 0.37423915433885206387815653913250097460*I -I 0.37423915433885206387815653913250097460 + 3.6720881244151097674232871083920323733*I] [1 -2.6720881244151097674232871083920323733 - 0.37423915433885206387815653913250097460*I I -0.37423915433885206387815653913250097460 + 1.6720881244151097674232871083920323733*I] [1 -2.6720881244151097674232871083920323733 + 0.37423915433885206387815653913250097460*I -I -0.37423915433885206387815653913250097460 - 1.6720881244151097674232871083920323733*I]
(matdet(C))^2
13568.000000000000000000000000000000000 + 0.E-34*I
A=[1,1/2*(1+I)*(sqrt(2+2*I)),I,I*1/2*(1+I)*(sqrt(2+2*I));1,1/2*(1-I)*(sqrt(2-2*I)),-I,-I*1/2*(1-I)*(sqrt(2-2*I));1,1/2*(1+I)*(-sqrt(2+2*I)),I,I*1/2*(1+I)*(-sqrt(2+2*I));1,1/2*(1-I)*(-sqrt(2-2*I)),-I,-I*1/2*(1-I)*(-sqrt(2-2*I))]
[1 0.45508986056222734130435775782246856962 + 1.0986841134678099660398011952406783786*I I -1.0986841134678099660398011952406783785 + 0.45508986056222734130435775782246856962*I] [1 0.45508986056222734130435775782246856962 - 1.0986841134678099660398011952406783785*I -I -1.0986841134678099660398011952406783785 - 0.45508986056222734130435775782246856962*I] [1 -0.45508986056222734130435775782246856962 - 1.0986841134678099660398011952406783786*I I 1.0986841134678099660398011952406783785 - 0.45508986056222734130435775782246856962*I] [1 -0.45508986056222734130435775782246856962 + 1.0986841134678099660398011952406783785*I -I 1.0986841134678099660398011952406783785 + 0.45508986056222734130435775782246856962*I]
(matdet(A))^2
511.99999999999999999999999999999999999 + 0.E-35*I
nfdisc(x^4-4*x^2+3)
12
B=[1,sqrt(2+I),I,I*sqrt(2+I);1,sqrt(2-I),-I,-I*sqrt(2-I);1,-sqrt(2+I),I,I*-sqrt(2+I);1,-sqrt(2-I),-I,-I*-sqrt(2-I)]
[1 1.4553466902253548081226618397096970699 + 0.34356074972251246413856574391455856847*I I -0.34356074972251246413856574391455856847 + 1.4553466902253548081226618397096970699*I] [1 1.4553466902253548081226618397096970699 - 0.34356074972251246413856574391455856847*I -I -0.34356074972251246413856574391455856847 - 1.4553466902253548081226618397096970699*I] [1 -1.4553466902253548081226618397096970699 - 0.34356074972251246413856574391455856847*I I 0.34356074972251246413856574391455856847 - 1.4553466902253548081226618397096970699*I] [1 -1.4553466902253548081226618397096970699 + 0.34356074972251246413856574391455856847*I -I 0.34356074972251246413856574391455856847 + 1.4553466902253548081226618397096970699*I]
(matdet(B))^2
1280.0000000000000000000000000000000000 - 6.728922305550388939 E-36*I
nfbasis(x^4-4*x^2+3)
[1, x, 1/2*x^2 + 1/2, 1/4*x^3 + 1/4*x^2 + 1/4*x + 1/4]
nfdisc(x^4-4*x^2+20)
1280
factor(1280)
[2 8] [5 1]
factor(8704)
[2 9] [17 1]
8704/4
2176
2176/9
2176/9
nfbasis(x^4-6*x^2+34)
[1, x, 1/5*x^2 + 2/5, 1/5*x^3 + 2/5*x]
factor(64)
[2 6]
2176/64
34
8704/16
544
factor(544)
[2 5] [17 1]
A=[1,I,sqrt(3+5*I),I*sqrt(3+5*I);1,-I,sqrt(3-5*I),-I*sqrt(3-5*I);1,I,-sqrt(3+5*I),I*-sqrt(3+5*I);1,-I,-sqrt(3-5*I),-I*-sqrt(3-5*I)]
[1 I 2.1013033925215678467731335152161665233 + 1.1897377641407581357639680612798653140*I -1.1897377641407581357639680612798653140 + 2.1013033925215678467731335152161665233*I] [1 -I 2.1013033925215678467731335152161665233 - 1.1897377641407581357639680612798653140*I -1.1897377641407581357639680612798653140 - 2.1013033925215678467731335152161665233*I] [1 I -2.1013033925215678467731335152161665233 - 1.1897377641407581357639680612798653140*I 1.1897377641407581357639680612798653140 - 2.1013033925215678467731335152161665233*I] [1 -I -2.1013033925215678467731335152161665233 + 1.1897377641407581357639680612798653140*I 1.1897377641407581357639680612798653140 + 2.1013033925215678467731335152161665233*I]
(matdet(A))^2
8703.9999999999999999999999999999999997 + 0.E-34*I
nfbasis(x^4-4*x^2+5)
[1, x, x^2, x^3]
factor(1280)
[2 8] [5 1]
nfdisc(x^4-8*x^2+20)
320
factor(320)
[2 6] [5 1]
factor(68)
[2 2] [17 1]
24*24
576
16*39
624
nfbasis(x^4-14*x^2+53)
[1, x, 1/2*x^2 + 1/2, 1/4*x^3 + 1/4*x^2 - 1/4*x - 1/4]
nfdisc(x^4-14*x^2+53)
3392
factor(3392)
[2 6] [53 1]
?euler phi
eulerphi(x): Euler's totient function of x.
eulerphi(4)
2
eulerphi(1)
1
eulerphi(2)
1
eulerphi(3)
2
nfbasis(x^4-34*x^2+650)
[1, x, 1/19*x^2 + 2/19, 1/95*x^3 + 21/95*x]
factor(272)
[2 4] [17 1]
nfdisc(x^4-38*x^2+650)
6656
factor(6656)
[2 9] [13 1]
factor(650)
[2 1] [5 2] [13 1]
factor(95)
[5 1] [19 1]
17^2
289
289+361
650
19^2
361
17*2
34
expand((x^2-17)^2)
*** unknown function or error in formal parameters: expand((x^2-17)^2) ^------------------
?expand
expand: unknown identifier
?factor
factor(x,{lim}): factorization of x. lim is optional and can be set whenever x is of (possibly recursive) rational type. If lim is set return partial factorization, using primes up to lim (up to primelimit if lim=0)
A=[1,-sqrt(8+2*I)+1/4*((3+I)*sqrt(8+2*I)+2),I,I*(-sqrt(8+2*I)+1/4*((3+I)*sqrt(8+2*I)+2));1,-sqrt(8-2*I)+1/4*((3-I)*sqrt(8-2*I)+2),-I,-I*(-sqrt(8-2*I)+1/4*((3-I)*sqrt(8-2*I)+2));1,sqrt(8+2*I)+1/4*((3+I)*-sqrt(8+2*I)+2),I, I*(sqrt(8+2*I)+1/4*((3+I)*-sqrt(8+2*I)+2));1,sqrt(8-2*I)+1/4*((3-I)*-sqrt(8-2*I)+2),-I,-1*I*(sqrt(8-2*I)+1/4*((3-I)*-sqrt(8-2*I)+2))]
[1 -0.30024259022012041915890982074952138854 + 0.62481053384382658687960444744285144400*I I -0.62481053384382658687960444744285144400 - 0.30024259022012041915890982074952138854*I] [1 -0.30024259022012041915890982074952138854 - 0.62481053384382658687960444744285144399*I -I -0.62481053384382658687960444744285144399 + 0.30024259022012041915890982074952138854*I] [1 1.3002425902201204191589098207495213885 - 0.62481053384382658687960444744285144400*I I 0.62481053384382658687960444744285144400 + 1.3002425902201204191589098207495213885*I] [1 1.3002425902201204191589098207495213885 + 0.62481053384382658687960444744285144399*I -I 0.62481053384382658687960444744285144399 - 1.3002425902201204191589098207495213885*I]
(matdet(A))^2
271.99999999999999999999999999999999999 - 3.877349896605722789 E-36*I
13568/3392
4
16^2
256
25^2
625
625+256
881
factor(881)
[881 1]
nfbasis(x^4-22*x^2+170)
[1, x, 1/7*x^2 + 3/7, 1/7*x^3 + 3/7*x]
nfdisc(x^4-22*x^2+170)
43520
factor(43520)
[2 9] [5 1] [17 1]
nextprime(17)
17
17^2
289
16^2
256
289+256
545
factor(545)
[5 1] [109 1]
nfbasis(x^4-6*x^2+25)
[1, x, 1/8*x^2 + 1/2*x - 3/8, 1/40*x^3 - 11/40*x + 1/2]
factor(8720)
[2 4] [5 1] [109 1]
factor(650)
[2 1] [5 2] [13 1]
factor(73)
[73 1]
nfbasis(x^4-16*x^2+280)
[1, x, 1/6*x^2 - 1/3, 1/12*x^3 + 1/3*x]
factor(1168)
[2 4] [73 1]
nfdisc(x^4-16*x^2+280)
645120
factor(645120)
[2 11] [3 2] [5 1] [7 1]
factor(280)
[2 3] [5 1] [7 1]
36*6+64
280
96+4
100
nfdisc(x^4-4*x^2+298)
2746368
factor(2746368)
[2 11] [3 2] [149 1]
factor(298)
[2 1] [149 1]
nfbasis(x^4-20*x^2+484)
[1, x, 1/16*x^2 + 1/2*x + 3/8, 1/352*x^3 - 21/176*x + 1/2]
factor(226)
[2 1] [113 1]
13^2
169
64*6
384
384+100
484
79/400
79/400
factor(79)
[79 1]
16^2
256
256+36
292
nfdisc(x^4-32*x^2+292)
1168
factor(1168)
[2 4] [73 1]
factor(292)
[2 2] [73 1]