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f(x) b
2i
=
c
2i
p
i
c
2i
∈ K, i = 1, 2, 3
p q deg p = 4
deg q = 5 deg b
2i
= 4i 4x + 5y = 4i
i = 1, 2, 3 y = 4y
1
y ≤ i − 1 y = i x = 0 i = 5
y = 0
f(x) = x
5
+ px + q D(f) = 4
4
p
5
+ 5
5
q
4
D =
P
d
kl
p
k
q
l
4k +5l = 20
D = d
1
p
5
+ d
2
q
4
p = 0, q = −1 D = −256 d
1
= 256 =
4
4
D p = 0, q = 1 D = 5
5
d
2
= 5
5
f
1
(x) = x
5
− x x
1
= 0, x
2
= i, x
3
= −1, x
4
=
−i, x
5
= 1 (i =
√
−1) ⇒ D(f
1
) = −256
G
0
(z) = z
6
+ 40z
5
+ 880z
4
+ 8960z
3
+ 44800z
2
+ 10854z + 102400. (4)
f
1
(x) = x
5
− x x
1
= 0, x
2
= i, x
3
=
−1, x
4
= −i, x
5
= 1 (i =
√
−1) h(x
1
, ..., x
5
) = −2i h
a
2
=
h
a
4
= h
a
5
= −2i, h
a
3
= 2 + 4i, h
a
6
= −4 + 2i
f(x) = x
5
+ px + q = 0
R(y) = (y
3
− 5py
2
+ 15p
2
y + 5p
3
)
2
− Dy
K
R(y) G(z) y =
1
4
z
R(y)
pq = 0
f(x) =
x
n
+ a
1
x
n−1
+ ···+ a
n−1
x + a
n
, a
i
∈ K α
1
, α
2
, ..., α
n
y(x) =
q( x)
r(x)
x = α
i
r(x) f(x)
(f(x), r(x)) = 1 g(y) = y
n
+b
1
y
n−1
+···+b
n
β
i
= y(α
i
), i = 1, ..., n
f(x) g(y)
f(x)
y(x) =
s(x)
t(x)
(f(x), t(x)) =
1 z(x) ≤ n − 1 (n = deg f(x))
z(α
i
) = y(α
i
) α
i
f(x)
(f(x), t(x)) = 1
u(x), v(x) u(x)f(x)+ v(x)t(x) = 1
v(α
i
) = 1/t(α
i
) f(α
i
) = 0 t(α
i
) 6= 0
s(α
i
)
t(α
i
)
=
Ëåììà 6. Åñëè f (x) íîðìàëüíûé ìíîãî÷ëåí ñòåïåíè 5, òî èìååì b2i =
c2i pi, ãäå c2i ∈ K, i = 1, 2, 3.
Äîêàçàòåëüñòâî. Êàê ìíîãî÷ëåíû îò êîðíåé p è q èìåþò deg p = 4 è
deg q = 5, à deg b2i = 4i (Ïîäñ÷èòàòü!). Êàæäîå óðàâíåíèå 4x + 5y = 4i äëÿ
i = 1, 2, 3 ìîæåò èìåòü öåëî÷èñëåííîå ðåøåíèå òîëüêî ïðè y = 4y1 . Ïðè
ýòîì y ≤ i − 1 (åñëè y = i, òî x = 0 è i = 5 ýòî íåâîçìîæíî). Ïîýòîìó
y = 0.
Ëåììà 7. Äëÿ f (x) = x5 + px + q èìååì D(f ) = 44 p5 + 55 q 4 .
Äîêàçàòåëüñòâî. Äîëæíû èìåòü D = P dkl pk ql , ãäå 4k + 5l = 20. Îòñþäà
D = d1 p5 + d2 q 4 . Òàê êàê ïðè p = 0, q = −1 èìååì D = −256, òî d1 = 256 =
44 . Çàòåì âû÷èñëèì D äëÿ p = 0, q = 1, ïîëó÷èì D = 55 , îòêóäà d2 = 55 .
Ëåììà 8. Äëÿ √ f1 (x) = x − x (êîðíè: x1 = 0, x2 = i, x3 = −1, x4 =
5
−i, x5 = 1 (i = −1) ⇒ D(f1 ) = −256) èìååì
G0 (z) = z 6 + 40z 5 + 880z 4 + 8960z 3 + 44800z 2 + 10854z + 102400. (4)
Äîêàçàòåëüñòâî. Êîðíÿìè
√ f1 (x) = x − x ÿâëÿþòñÿ x1 = 0, x2 = i, xa 3 =
5
−1, x4 = −i, x5 = 1 (i = −1). Îòñþäà èìååì h(x1 , ..., x5 ) = −2i è h = 2
ha = ha = −2i, ha = 2 + 4i, ha = −4 + 2i. Ýòî âëå÷åò (4).
4 5 3 6
Ïðåäëîæåíèå 19. Íîðìàëüíîå óðàâíåíèå f (x) = x5 + px + q = 0 ðàç-
ðåøèìî â ðàäèêàëàõ òîãäà è òîëüêî òîãäà, êîãäà ìíîãî÷ëåí
R(y) = (y 3 − 5py 2 + 15p2 y + 5p3 )2 − Dy
èìååò êîðåíü â K .
Äîêàçàòåëüñòâî. R(y) ïîëó÷àåòñÿ èç G(z) çàìåíîé y = 41 z. Ïîýòîìó, åñëè
R(y) íå èìååò êðàòíûõ êîðíåé, òî óòâåðæäåíèå ñëåäóåò èç ïðåäûäóùåãî,
åñëè æå èìååò, òî pq = 0.
12. Ïðåîáðàçîâàíèå ×èðíãàóçåíà
Îïðåäåëåíèå ïðåîáðàçîâàíèÿ ×èðíãàóçåíà. Ïóñòü çàäàíû ìíîãî÷ëåí f (x) =
xn + a1 xn−1 + · · · + an−1 x + an , ai ∈ K, ñ êîðíÿìè è ðàöèîíàëü-
α1 , α2 , ..., αn
íàÿ ôóíêöèÿ , îïðåäåëåííàÿ ïðè êàæäîì çíà÷åíèè
q(x)
y(x) = r(x) x = αi(èíà-
÷å ãîâîðÿ, ñî çíàìåíàòåëåì , íå èìåþùèì îáùèõ êîðíåé ñ , ò.å. ñ
r(x) f (x)
(f (x), r(x)) = 1). Òðåáóåòñÿ ïîñòðîèòü ìíîãî÷ëåí g(y) = y n +b1 y n−1 +· · ·+bn,
êîðíÿìè êîòîðîãî ÿâëÿþòñÿ βi = y(αi ), i = 1, ..., n . Ïåðåõîä îò ïåðâîãî
ìíîãî÷ëåíà f (x) êî âòîðîìó g(y)íàçûâàåòñÿ ïðåîáðàçîâàíèåì ×èðíãàóçåíà
(ìíîãî÷ëåíà ). f (x)
Ïðåæäå âñåãî, îòìåòèì ñëåäóþùóþ ëåììó:
Ëåììà 1. Äëÿ âñÿêîé ðàöèîíàëüíîé ôóíêöèè y(x) = t(x) ñ (f (x), t(x)) =
s(x)
1 ñóùåñòâóåò ìíîãî÷ëåí z(x) ñòåïåíè ≤ n − 1 (n = deg f (x)) òàêîé, ÷òî
z(αi ) = y(αi ) äëÿ âñåõ êîðíåé αi ìíîãî÷ëåíà f (x).
Äîêàçàòåëüñòâî. Òàê êàê (f (x), t(x)) = 1, òî ñóùåñòâóþò ìíîãî÷ëåíû
u(x), v(x), äëÿ êîòîðûõ èìååò ìåñòî u(x)f (x) + v(x)t(x) = 1. Èç ýòîãî ðàâåí-
ñòâà èìååì v(αi ) = 1/t(αi ) (ò.ê. f (αi ) = 0, à t(αi ) 6= 0). Ñëåäîâàòåëüíî, s(αi)
t(α ) =
i
33
