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Àëãåáðà II, îñåííèé ñåìåñòð 2014 ã.
Çàäà÷è äëÿ ñåìèíàðà 5.
ôàêóëüòåò ìàòåìàòèêè, ÍÈÓ ÂØÝ
Çàäà÷à 1
(Êðèòåðèé ؼíåìàííà). (à) Ìíîãî÷ëåí f (x) ∈ Z[x] èìååò âèä
f (x) = (x − a)n + pF (x)
äëÿ íåêîòîðûõ a, p ∈ Z, ãäå F (x) ∈ Z[x] ìíîãî÷ëåí ñòåïåíè íå âûøå n, à p ïðîñòîå. Äîêàæèòå, ÷òî åñëè F (a) 6= 0 (mod p), òî f (x) íåïðèâîäèì â Z[x].
(á) Âûâåäèòå êðèòåðèé Ýéçåíøòåéíà èç êðèòåðèÿ ؼíåìàííà.
(â) Íåïðèâîäèì ëè â Z[x] ìíîãî÷ëåí
x5 − 2x4 + 13x3 − 7x2 + 8x + 2?
(ã) Ïðèâåäèòå êîíòðïðèìåð ê êðèòåðèþ ؼíåìàííà, åñëè íå òðåáîâàòü deg(F ) ≤ n.
(à) Ñôîðìóëèðóéòå êðèòåðèé ؼíåìàííàÝéçåíøòåéíà äëÿ êîëüöà ìíîãî÷ëåíîâ R[x], ãäå R ïðîèçâîëüíàÿ îáëàñòü öåëîñòíîñòè.
(á) Íåïðèâîäèì ëè â C[x, y] ìíîãî÷ëåí
Çàäà÷à 2.
x3 y 2 + x4 + 4x3 y + 4x3 + y + 2?
Çàäà÷à 3.
(1) Z[e
2πi
3
Êàêèå èç ñëåäóþùèõ êîëåö (à) åâêëèäîâû (á) ôàêòîðèàëüíû?
√
√
]; (2) R[x, y]; (3) Z[x]; (4) Z[ −2]; (5) Z[ 5].
Ïóñòü
d öåëîå ÷èñëî, íå ÿâëÿþùååñÿ ïîëíûì êâàäðàòîì. Íàçîâ¼ì ýëå√
ìåíò α ∈ Q( d) öåëûì àëãåáðàè÷åñêèì ÷èñëîì, åñëè α ÿâëÿåòñÿ êîðíåì ìíîãî÷ëåíà
x2 + px + q äëÿ íåêîòîðûõ p, q ∈ Z.
√
(à) Äîêàæèòå, ÷òî âñå öåëûå àëãåáðàè÷åñêèå ýëåìåíòû ïîëÿ
Q(
d) îáðàçóþò êîëü√
öî (îíî íàçûâàåòñÿ êîëüöîì öåëûõ êâàäðàòè÷íîãî ïîëÿ Q( d)).
(á) Äîêàæèòå, ÷òî ïðè d = −1 êîëüöî öåëûõ ñîâïàäàåò ñ êîëüöîì Z[i] öåëûõ ÷èñåë
2πi
Ãàóññà, à ïðè d = −3 ñîâïàäàåò ñ êîëüöîì Z[e 3 ] öåëûõ
√ ÷èñåë Ýéçåíøòåéíà.
(â) Íàéäèòå êîëüöî öåëûõ êâàäðàòè÷íîãî ïîëÿ Q( d) äëÿ ïðîèçâîëüíîãî d.
Çàäà÷à 4.
Äîêàæèòå, ÷òî ïðè d = 1, 2, 3, 7, 11 êîëüöî √
öåëûõ êâàäðàòè÷íîãî ïîëÿ
√
Q( −d) áóäåò åâêëèäîâûì îòíîñèòåëüíî íîðìû N (a + b d) = a2 + db2 .
Çàäà÷à 5.
2
Problem 1
(Sch
onemann criterion). (a) A polynomial f (x) ∈ Z[x] can be written as
f (x) = (x − a)n + pF (x)
for some a, p ∈ Z, where F (x) ∈ Z[x] is a polynomial of degree at most n, and p is prime.
Prove that if F (a) 6= 0 (mod p), then f (x) is irreducible in Z[x].
(b) Deduce the Eisenstein criterion from the Schonemann criterion.
(c) Is the following polynomial irreducible in Z[x]
x5 − 2x4 + 13x3 − 7x2 + 8x + 2?
(d) Give a counterexample to the Schonemann criterion when the condition deg(F ) ≤ n
is omitted.
(a) Formulate the EisensteinSchonemann criterion for the ring R[x], where
R is an arbitrary integral domain.
(b) Is the following polynomial irreducible in C[x, y]
Problem 2.
x3 y 2 + x4 + 4x3 y + 4x3 + y + 2?
Problem 3.
Domains?
2πi
(1) Z[e 3 ];
Which of the following rings are (a) Euclidean (b) Unique Factorization
√
(5) Z[ 5].
√
Problem 4. Let d be a square free integer. An element α ∈ Q( d) is called algebraic
integer if α is a root of x2 + px + q for some
√ p, q ∈ Z.
(a) Prove that all algebraic integers
√ in Q( d) form a ring (it is called the ring of integers
of the quadratic number eld Q( d)).
(b) Prove that if d = −1 the the ring of integers coincide with the ring Z[i] of Gaussian
2πi
integers, and if d = −3 then it coincides with the ring Z[e 3 ] of the Eisenstein integers
(also known as Eulerian integers). √
(c) Find the ring of integers of Q( d) for all d.
√
Problem 5. Show that if d = 1, 2, 3, 7, 11 then the ring of integers of Q( −d) is
√
Euclidean with respect to the norm N (a + b d) = a2 + db2 .
(2) R[x, y];
√
(3) Z[x]; (4) Z[ −2];
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