2 1. POLYNOMIAL S I N ON E VARIABL E
from th e resultan t (denote d Res
x
an d discusse d i n Chapte r 4 ) o f th e polynomia l
p{x) an d it s firs t derivativ e p'{x) a s follows :
discrx{p{x)) = Res
x(p(x),p,(x)).
This i s a n irreducibl e polynomia l i n th e coefficient s ao,ai,.. . , a^. I t follow s fro m
Sylvester's matri x formul a fo r the resultant tha t th e discriminant i s a homogeneou s
polynomial o f degree 2d 2. Her e i s the discriminan t o f a quartic :
f : = a 4 * x~ 4 + a 3 * x~ 3 + a 2 * x~ 2 + a l * x + a O :
lprint(resultant(f,diff(f,x),x)/a4) ;
-192*a4~2*a0~2*a3*al-6*a4*a0*a3~2*al~2+144*a4*a0~2*a2*a3~2
+144*a4~2*a0*a2*al~2+18*a4*a3*al~3*a2+a2~2*a3~2*al~2
-4*a2~3*a3~2*a0+256*a4~3*a0~3-27*a4~2*al~4-128*a4~2*a0~2*a2~2
-4*a3~3*al~3+16*a4*a2~4*a0-4*a4*a2~3*al~2-27*a3~4*a0~2
-80*a4*a3*al*a2~2*a0+18*a3~3*al*a2*a0
This sexti c i s the determinan t o f the followin g 7 x 7-matri x divide d b y a4 :
with(linalg) :
sylvester(f,diff(f,x),x) ;
[ a4
[ 0
[ 0
[4 a4
[ 0
[ 0
[ 0
a3
a4
0
3 a3
4 a4
0
0
a2
a3
a4
2 a2
3 a3
4 a4
0
al
a2
a3
al
2 a2
3 a3
4 a4
aO
al
a2
0
al
2 a2
3 a3
0
aO
al
0
0
al
2 a2
o
i—
i
i—
i
o ]
]
aO]
]
0 ]
]
0 ]
]
0 ]
]
al]
Galois theory tells us that ther e is no general formula whic h expresses the root s
o£p(x) i n radicals i f d 5. Fo r specifi c instance s wit h d not to o big , sa y d 10, it
is possible t o comput e th e Galoi s grou p o f p(x) ove r Q. Occasionally , on e i s luck y
and th e Galoi s grou p i s solvable, i n whic h cas e mapl e ha s a chanc e o f finding th e
solution o f p(x) 0 in term s o f radicals .
f : = x~ 6 + 3*x~ 5 + 6*x~ 4 + 7*x~ 3 + 5*x~ 2 + 2* x + 1:
galois(f) ;
"6T11", {"[2~3]S(3) \ " 2 w r S ( 3 ) \
H
2S_4(6)"},
{"(2 4 6)( 1 3 5)" , "( 1 5 ) (2 4)" , "( 3 6)" }
solve(f,x ) [1];
1/2 1/3
1/12 (- 6 (108 + 12 6 9 )
48,
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