Введение в теорию Галуа. Ермолаев Ю.Б. - 17 стр.

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K c
1
, ..., c
n
K
h
k
= h
k
(x
1
, ..., x
k
) = c
1
x
1
+···+c
k
x
k
(h
a
k
)
0
= (h
b
k
)
0
x
a
i
= x
b
i
, i = 1, ..., k (h
k
)
0
= c
1
α
1
+ ··· + c
k
α
k
a, b S
n
h
k
= h
k1
+ c
k
x
k
(c
1
6= 0) h
a
k
= h
a
k1
+
c
k
x
i
, h
b
k
= h
b
k1
+ c
k
x
j
, i 6= j x
a
k
= x
i
, x
b
k
= x
j
h
a
k
h
b
k
=
h
a
k1
h
b
k1
+ c
k
(α
i
α
j
) 6= 0 c
k
6=
h
a
k1
h
b
k1
α
i
α
j
a, b S
n
c
i
6= c
j
i 6= j h
h
a
0
= h
b
0
a = b (a, b S
n
)
h
a
0
K(α
1
, ..., α
n
)
h
a
c
i
h
G = {b
1
, ..., b
m
} h = h
n
ϕ(t) = (th
b
1
) ···(th
b
m
) t
0
K g
a
i
6= g
a
j
i 6= j g = ϕ(t
0
) a
1
, ..., a
s
G g G
G(z) = (z g
a
1
0
) ···(z g
a
s
0
)
G(z) g
a
i
0
g
n = 3 S
3
E = {(1)}, C
2
= {(1), (12)}, C
0
2
=
{(1), (13)}, C
00
2
= {(1), (23)}, A
3
= {(1), (123), (132)}, S
3
f(x) = x
n
σ
1
x
n1
+
··· + (1)
n1
σ
n1
x + (1)
n
σ
n
, σ
i
K α
1
, α
2
, ..., α
n
y = a
0
+ a
1
x + ··· + a
n1
x
n1
, a
i
K
g(y) = y
n
Σ
1
y
n1
+ ··· + (1)
n
Σ
n
β
i
= y(α
i
), i = 1, ..., n
f(x) g(y)
f(x) g(y)
f(x)
f(x) = x
3
2x + 3, y = 1 x + x
2
y
i
y = 1 x + x
2
, y
2
=
7 9x + 5x
2
, y
3
= 49 59x + 31x
2
(mod f(x)) S
1
= β
1
+ β
2
+
β
3
= 3 s
1
+ s
2
, S
2
= 21 9s
1
+ 5s
2
, S
3
= 147 59s
1
+ 31s
2
σ
1
= 0, σ
2
= 2, σ
3
= 3 s
1
= 0, s
2
= 4 S
1
= 7, S
2
= 41, S
3
=
271 Σ
1
= 7, Σ
2
= 4, Σ
3
= 4 g(y) = y
3
7y
2
+ 4y 4
1, y, y
2
, y
3
(mod f(x))
y = 1 x + x
2
, xy =
3 + 3x x
2
, xy
2
= 3 5x + 3x
2
(mod f(x))
(1, x, x
2
)
1 y 1 1
3 3 y 1
3 5 3 y
= 0.
g(y) K
      Ëåììà 6. Åñëè ïîëå K áåñêîíå÷íî, òî ñóùåñòâóþò c1 , ..., cn ∈ K òàêèå,
÷òî äëÿ êàæäîãî hk = hk (x1 , ..., xk ) = c1 x1 +· · ·+ck xk èìååì (hak )0 = (hbk )0 ⇔
xai = xbi , i = 1, ..., k , ãäå (hk )0 = c1 α1 + · · · + ck αk è a, b ∈ Sn .
      Äîêàçàòåëüñòâî. Èìååì hk = hk−1 + ck xk (c1 6= 0); ïóñòü hak = hak−1 +
ck xi , hbk = hbk−1 + ck xj , i 6= j (ò.å. xak = xi , xbk = xj ). Òîãäà hak − hbk =
                                                                äëÿ ëþáûõ a, b ∈ Sn . 
                                                a     b
                                                    h    −h
hak−1 − hbk−1 + ck (αi − αj ) 6= 0 ⇔ ck 6= − α −α
                                                k−1   k−1


      Çàìå÷àíèå 2. Óñëîâèå: ci 6= cj ïðè i 6= j â îïðåäåëåíèè h íåäîñòàòî÷íî
                                                  i   j



äëÿ òîãî, ÷òîáû ha0 = hb0 ⇔ a = b (a, b ∈ Sn ) (ïî ñðàâíåíèþ ñ äîêàçàòåëü-
ñòâîì ïðåäëîæåíèÿ 1 çäåñü ha0  ýëåìåíòû ïîëÿ K(α1 , ..., αn ), òîãäà êàê òàì
ha  ìíîãî÷ëåíû). Ïîýòîìó çäåñü äëÿ âûáîðà ci ïðèõîäèòñÿ ïðèìåíÿòü áî-
ëåå òîíêîå ðàññóæäåíèå. Íî ïîñëå òîãî, êàê óäàåòñÿ íàéòè íóæíîå h, äîâîäû
äîêàçàòåëüñòâà ïðåäëîæåíèÿ 1 ñîõðàíÿþò ñâîþ ñèëó.
      Äîêàçàòåëüñòâî ïðåäëîæåíèÿ 4. Ïóñòü G = {b1 , ..., bm }, h = hn (ñì. ëåì-
ìó 6) è ϕ(t) = (t − hb ) · · · (t − hb ). Åñëè t0 ∈ K ïîäîáðàíî òàê, ÷òî ga 6= ga
                      1            m                                         i     j

ïðè i 6= j , ãäå g = ϕ(t0 ), à a1 , ..., as  ïîëíàÿ ñèñòåìà ïðåäñòàâèòåëåé äëÿ
G, òî g  îïðåäåëÿþùèé ìíîãî÷ëåí äëÿ G, ñîîòâåòñòâóþùèé ìíîãî÷ëåí
G(z) = (z − g0a ) · · · (z − g0a ) êîòîðîãî áóäåò èñêîìûì. Äåéñòâèòåëüíî, êîð-
                1            s


íÿìè G(z) ÿâëÿþòñÿ ýëåìåíòû g0a , êîòîðûå ïî ïîñòðîåíèþ g âñå ðàçëè÷íû.
                                       i



      Ïðèìåðû. Ñëó÷àé n = 3. Ïîäãðóïïû S3 : E = {(1)}, C2 = {(1), (12)}, C20 =
{(1), (13)}, C200 = {(1), (23)}, A3 = {(1), (123), (132)}, S3 .
      Ïðåîáðàçîâàíèå ×èðíãàóçåíà. Ïóñòü çàäàíû ìíîãî÷ëåí f (x) = xn −σ1 xn−1 +
· · · + (−1)n−1 σn−1 x + (−1)n σn , σi ∈ K , ñî ñâîèìè êîðíÿìè α1 , α2 , ..., αn è
ïðåîáðàçîâàíèå y = a0 + a1 x + · · · + an−1 xn−1 , ai ∈ K .
      Òðåáóåòñÿ ïîñòðîèòü ìíîãî÷ëåí g(y) = yn − Σ1 yn−1 + · · · + (−1)n Σn ,
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)). Ðàññìîòðèì äâà ìåòîäà ïîñòðîåíèÿ ìíîãî÷ëåíà g(y) äëÿ
äàííîãî f (x) íà êîíêðåòíîì ïðèìåðå.
      Ïóñòü f (x) = x3 − 2x + 3, y = 1 − x + x2 .
      1) Íàéäåì ñòåïåííûå ñóììû äëÿ âåëå÷èí yi . Èìååì y = 1 − x + x2 , y2 =
7 − 9x + 5x2 , y 3 = 49 − 59x + 31x2 (mod f (x)). Ïîýòîìó S1 = β1 + β2 +
β3 = 3 − s1 + s2 , S2 = 21 − 9s1 + 5s2 , S3 = 147 − 59s1 + 31s2 ; çàòåì, ò.ê.
σ1 = 0, σ2 = −2, σ3 = −3, òî s1 = 0, s2 = 4; îòñþäà S1 = 7, S2 = 41, S3 =
271 ⇒ Σ1 = 7, Σ2 = 4, Σ3 = 4. Ñëåäîâàòåëüíî, g(y) = y 3 − 7y 2 + 4y − 4.
      Çàìå÷àíèå. Ìîæíî òàêæå (è âîçìîæíî ýòî ïðîùå) íàéòè çàâèñèìîñòü
ìåæäó 1, y, y2 , y3 (mod f (x)).
      2) Ðàññìîòðèì ñëåäóþùóþ ñèñòåìó ìíîãî÷ëåíîâ: y = 1 − x + x2 , xy =
−3 + 3x − x2 , xy 2 = 3 − 5x + 3x2 (mod f (x)). Òàê êàê ýòà ñèñòåìà äîëæíà
èìåòü ðåøåíèå îòíîñèòåëüíî (1, x, x2 ), òî äîëæíîèìåòü ìåñòî ðàâåíñòâî
                             1−y        −1       1
                              −3       3−y       −1       = 0.
                               3        −5      3−y
Ýòî è åñòü g(y) (ñ òî÷íîñòüþ äî ìíîæèòåëÿ èç K ).
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