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algebraically solvable∗
pahio†
2013-03-22 0:32:20
An equation
xn + a1 xn−1 + . . . + an = 0,
(1)
with coefficients aj in a field K, is algebraically solvable, if some of its roots may
be expressed with the elements of K by using rational operations (addition,
subtraction, multiplication, division) and root extractions. I.e., a root of (1)
is in a field K(ξ1 , ξ2 , . . . , ξm ) which is obtained of K by adjoining to it in
succession certain suitable radicals ξ1 , ξ2 , . . . , ξm . Each radical may be contain
under the root sign one or more of the previous radicals,

√

ξ1 = p1 r1 ,


p

p2


r2 (ξ1 ),
ξ2 = p
p3
ξ3 =
r3 (ξ1 , ξ2 ),



···
···


p

ξ = pm
rm (ξ1 , ξ2 , . . . , ξm−1 ),
m
where generally rk (ξ1 , ξ2 , . . . , ξk−1 ) is an element of the field K(ξ1 , ξ2 , . . . , ξk−1 )
but no pk ’th power of an element of this field. Because of the formula
q
√
j √
jk
k
r=
r
one can, without hurting the generality, suppose that the indices p1 , p2 , . . . , pm
are prime numbers.
Example. Cardano’s formulae show that all roots of the cubic equation
y 3 + py + q = 0 are in the algebraic number field which is obtained by adjoining
to the field Q(p, q) successively the radicals
r r
√
q 2
p 3
q
ξ1 =
+
, ξ2 = 3 − +ξ1 , ξ3 = −3.
2
3
2
∗ hAlgebraicallySolvablei
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In fact, as we consider also the equation (4), the roots may be expressed as

p
y1 = ξ2 −



3ξ
2


−1+ξ3
−1−ξ3 p
y2 =
· ξ2 −
·
2
2
3ξ2




y3 = −1−ξ3 · ξ2 − −1+ξ3 · p
2
2
3ξ2
References
[1] K. Väisälä: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No.
17. Kustannusosakeyhtiö Otava, Helsinki (1950).
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