Bordeaux 1 University MHT 933 - master 2
Mathematics 2009-2010
Exercises for Chapter 3
Exercise 1– [Wilson]
1. LetFq be a finite field. Prove that Y
α∈F×q
α=−1,
and in particular that if p is prime (p−1)!≡ −1 modp.
2. Conversely, show that n ≥ 2 being an integer such that (n − 1)! ≡
−1 mod n, then n is prime.
Exercise 2– [Yun’s algorithm]
Yun’s algorithm is an efficient algorithm that computes not only the square- free part but the full squarefree decomposition of a polynomial: ifk is a field and if P ∈ k[x] is nonconstant and monic, the squarefree decomposition of P is
P =f1f22· · ·fmm
where the fi are monic squarefree pairwise coprime polynomials andfm 6= 1.
With the same notation the squarefree part of P is f1· · ·fm. For instance, the squarefree decomposition of x4(x+ 1)2(x−1)2(x2 + 1)2(x2 +x+ 1) in Q[X] is (x2+x+ 1)(x4−1)2(1)3(x)4. Thusfi is the product of the irreducible monic polynomials that divide P exactlyi times.
1. We first suppose that k has characteristic zero. Let f1, . . . , fm ∈k[X] be monic squarefree and pairewise coprime polynomials. Let us put
f =f1· · ·fm and g = X
1≤i≤m
cifi0f fi, for some constants ci ∈k. Show that
gcd(f, g−cf0) = Y
cj=c
fj.
2. Consider the following algorithm. Givenf ∈k[X] monic of degreen ≥1, where k has still characteristic zero:
1. Put i= 1, u= gcd(f, f0),v1 =f /u and w1 =f0/u 2. Whilevi 6= 1 do
hi = gcd(vi, wi−v0i),vi+1 =vi/hi, wi+1 = (wi−vi0)/hi, i=i+ 1 End do
3. Return (h1, . . . , hi−1)
Show that the finalhj give the squarefree decomposition off: f =h1h22· · ·hi−1i−1. 3. Show that it uses ˜O(n) operations in k.
4. Suppose f =abc2d4 for monic distinct irreducible polynomials a, b, c, d∈ k[X]. Follow in detail the different steps of the algorithm applied to this f. 5. Suppose from now on that k = Fq with q =p` and that f ∈ k[X] monic and nonconstant has squarefree decomposition f1f22· · ·fmm. Show that the previous algorithm returns the correct answer if m < p.
6. Letm ≥p. Show that the algorithm when applied to f computes hi = Y
j≡imodp
fj
for 1≤i < p and hi = 1 for i≥p.
7. Modify the algorithm so as to work also for m≥p.
8. Show that the modified algorithm takes ˜O(`n) operations inFq.
9. Trace the algorithm withf =ab2c2d6e8 ∈F2[X] wherea, b, c, d, e∈F2[X]
are irreducible and pairwise coprime.
Exercise 3– [Probability to be squarefree]
We want to prove that for a prime power q=pf and a positive integern ≥2, the probability for a random polynomial inFq[X] of degreento be squarefree is 1−1/q (Carlitz, 1932).
Let sn denote the number of monic squarefree polynomials of degree n in Fq[X]. Thens0 = 1 ands1 =q.
1. Prove the recursive formula X
0≤2k≤n
qksn−2k =qn.
2. Prove thatsn=qn−qn−1 if n≥2 and conclude.
Exercise 4– [Irreducible factors of Xn−1 in Fq[X]]
Let q=pf a prime power and n≥2 coprime to q. We define an equivalence relation ∼ onZ/nZ by
x∼y⇐⇒there exists l ∈Z such thaty =xql.
Let S1, . . . , Sr be the distinct equivalence classes of Z/nZ with respect to∼ and
bi =X
j∈Si
xj ∈Fq[X] for 1≤i≤r.
1. Consider B ⊆Fq[X]/hTithe Berlekamp algebra associated to the polyno- mialT =Xn−1. Show that a basis ofB is given byb1 modT,. . . , br modT. 2. Give a probabilistic algorithm which computes all irreducible factors of Xn−1 in Fq[X] using an expected number of ˜O(nlogq) field operations in Fq (Prange, 1959).
Exercise 5– [Irreducibility in Fq[X]] Let q=pf a prime power.
1. For any d ≥ 1, show that Xqd −X ∈ Fq[X] is the product of all monic irreducible polynomials in Fq[X] whose degree divides d.
2. Letf ∈Fq[X] of degree n≥1. Show that f is irreducible if and only if (i) f divides Xqn−X;
(ii) gcd(Xqn/t −X, f) = 1 for all prime divisor of n.
3. Assume that we know the prime divisors of n. Write an algorithm which tests the irreducibility of f ∈ Fq[X] using ˜O(n(ω+1)/2 +nlogq) operations in Fq. We shall admit here that if R is a commutative unitary ring and if f, g, h ∈ R[X] with f 6= 0 monic, deg g, deg h < deg f =n, there exists an algorithm which computes g(h) modf with O(n(ω+1)/2) operations inR.
4. LetI(n, q) be the number of monic irreducible polynomials of degree n in Fq[X]. Prove that
qn−2qn/2
n ≤I(n, q)≤ qn n .
5. Prove that if n is prime, there are exactly (qn − q)/n distinct monic irreducible polynomials of degree ninFq[X] and find a simple formula when n is a prime power.
6. Show that, using our previous algorithm, we can find a uniformly random
irreducible polynomial of degree n in Fq[X] using an expected number of O(n˜ (ω+3)/2+n2logq) operations inFq.
7. Consider now the following algorithm (Ben–Or’s generation of a uniformly random monic irreducible polynomial of degree n inFq[X]).
1. randomly choose a monic polynomial f ∈Fq[X] of degree n;
2. for i from 1 to bn/2c put gi =gcd(Xqi −X, f) and if gi 6= 1 then goto 1;
3. return f.
Show that Ben–Or’s algorithm works correctly as specified and takes an ex- pected number of ˜O(n2logq) operations inFq.
Exercise 6–[Gauss Lemma, Eisenstein criterion, cyclotomic poly- nomials]
1. LetRbe a UFD andP, Q∈R[X]. Show that cont(P Q) =cont(P)cont(Q).
2. LetP ∈R[X] non constant and K the field of fractions of R. Show that P is irreducible inR[X] if and only if it is primitive and irreducible inK[X].
3. Show thatR[X] is also a UFD and describe the primes of R[X].
4. Let P = Pn
i=0aiXi ∈ Z[X]. Suppose that there exists a prime p such that
p-an, p|ai for 0≤i≤n−1 and p2 -a0. Show that P is irreducible in Q[X].
5. Letp be a prime and Φp = (Xp −1)/(X−1) = 1 +X+· · ·+Xp−1 the p-th cyclotomic polynomial. By a suitable variable changing, prove that Φp is irreducible in Q[X].
6. Letn ≥1 and Φn the n-th cyclotomic polynomial defined by Φn = Y
ζ∈Ωn
(X−ζ),
where Ωn is the set of primitive n-th roots of 1. Show that this definition is compatible with the previous one in the case n prime. Then prove that Φn∈Z[X] and is irreducible in Q[X] for every n.
7. Let n = pq where p and q are distinct odd primes. Prove that the n-th cyclotomic polynomial Φn (which is irreducible in Q[X]) splits modulo any prime into at least two factors.
Exercise 7– [Elementary approaches]
1. Let P = X10 +X + 1. Using Berlekamp algorithm, we find that the factorizations of P into monic irreducible factors over Fp for p = 2, 3,5 respectively are:
P ≡ (X3+X+ 1)(X7 +X5+X4+X3+ 1) mod 2
P ≡ (X−1)(X3−X2−X−1)(X6−X5+X4−X3+X+ 1) mod 3 P ≡ (X2−X+ 2)(X8+X7 −X6 + 2X5−X4+ 2X2 + 2X−2) mod 5.
What can we say of P inQ[X] ?
2. Letp and q be two odd primes. We assume that q≡2 mod 3 and that p is a primitive root modulo q. Show that the polynomial
P =Xq+1−X+p is irreducible in Q[X].
3. Show that this criterion does not work withP =X4+ 1. More precisely, prove that P splits modulo p into 4 linear factors if p = 2 or p ≡ 1 mod 8, into two irreducible quadratic factors otherwise, and that P is irreducible in Q[X]. Hint: make use of the quadratic reprocity law.
4. Consider now the polynomial
P =X6−6X4−2X3−7X2+ 6X+ 1.
Show that ifQis a factor ofP of degree less or equal to 3 then its coefficients are bounded in absolute value by 23.
5. Berlekamp algorithm shows that over Z/47Z we have
P = (X−22)(X−13)(X−12)(X+ 12)(X2 −12X−4),
where the five factors are irreducible in Z/47Z[X]. Thanks to this decompo- sition, find the factorization of P into a product of monic irreducible poly- nomials in Q[X].
Exercise 8– [Swinnerton-Dyer polynomials]
Recall that the n-th Swinnerton-Dyer polynomial is defined by Pn=Y
(X±√ 2±√
3±√
5± · · · ±√ pn),
where pn is the n-th prime number and where the product runs over all 2n possible combinations of sign + and − signs.
1. Prove thatPn∈Z[X] for every n≥1.
2. Factor P2 modulo 2, 3 and 5. and prove that P2 is irreducible.
3. Prove that Pn is irreducible in Q[X] for every n ≥ 1. Hint: consider K =Q(√
2, . . . ,√
pn) and use Galois theory.
4. Let p be a prime number and n ≥ 1. Prove that all irreducible factors modulo p of Pn have degree at most 2 and if pdoes not divide the discrimi- nant of Pn, they are either all linear or all quadratic.
Exercise 9– [Linear Hensel step]
Let p ∈ Z>1. In Hensel step algorithm, we replace sg+ht ≡ 1 modm by sg+ht ≡ 1 modp. In step 1 we perform computations modulo mp instead of m2 and step 2 is omitted completely.
1. Prove that the output specifications forf,g∗ andh∗ hold ifm2 is replaced by pm.
2. Letf ∈Z[X] andg, h, r, s ∈Z[X] such that f ≡ghmodpand sg+ht≡ 1 mod p. Using the previous linear Hensel step, write an algorithm which computes a factorization of f modulo pl for some l≥1.
3. Compare the complexity of this new algorithm to the complexity of the algorithm which makes use of the usual quadratic Hensel step.
Exercise 10– [Hensel lifting]
Let f =X15−1∈Z[X].
1. Take a nontrivial factorizationf ≡ghmod 2 with g, h∈Z[X] monic and of degree at least 2.
2. Compute g∗ and h∗ ∈Z[X] such that
f ≡g∗h∗ mod 16, degg∗ = degg, g∗ ≡g mod 2.
3. Can you guess some factors off inZ[X]?
Exercise 11– [Hensel lifting]
Let f = 14X4+ 15X3+ 42X2+ 3X+ 1∈Z[X].
1. Find a suitable primepsuch thatf modpis squarefree and has degree 4.
2. Compute the irreducible factorization of f mod p in Fp[X]. Choose
two factors g, h ∈ Z[X] that are coprime modulo p such that h is monic and irreducible modulo p and f ≡ ghmodp. Determine s, t ∈ Z[X] with sg+th ≡1 modp.
3. Execute two Hensel steps to obtain a factorization f ≡g∗h∗ modp4 with g ≡g∗ modpandh≡h∗ modp. Can you derive a factorization off inQ[X]
from it?
Exercise 12– [Examples in number fields]
Let us see here some illustrations of the algorithm (seen in course) which gives the factorization of a polynomial over a number field. For this, if you are lazy or tired, you can obtain factorizations in Q[X] with the help of a computer.
1. Factor
P =x3−(7−3√
2)X+ 5√ 2−6
into a product of monic irreducible polynomials ofK[X] where K =Q(√ 2).
2. Same question withK =Q(√
−3) and P =X3+ 1−5√
−3
2 X2 +3−5√
−3
2 X−1−5√
−3.
3. Same question with K = Q(α) where α is a root of X3 −X2 −9X+ 8 and
P =X3−(6α−9)X2+ (5α2−33α+ 13)X−5α+ 30.
