is the factorization of f(X) modulo it, where each q>vCX)EAE[X] is chosen so that it does not divide f(x). Denote by m; the degree of q>v(X). Then f(X) = ] (X) .
r
FACTORIZATION OVER LOCAL FIELDS AND THE IRREDUCIBILITY OF GENERALIZED DIFFERENCE POLYNOMIALS S. D. COHEN, A. MOV AHHEDI
•.
AND
A. SALINIER
Let E be a local field, i.e., a field which is complete with respect to a rank one discrete valuation v (we do not require any finiteness condition on the residue class field of E). Let I(x) be a polynomial in one variable, with coefficients in E. It is well known [4, 6, 9, 11, 13] that the Newton polygon method allows us to gather information about the factorization of I(x). This method consists of attaching to each side S of a Newton polygon of I(x) a factor (not necessarily irreducible) of I(x), the degree of which is the length of the horizontal projection of S. We can further refine this factorization by using a polynomial associated with a side of the Newton polygon (see Definition 1.4). To our knowledge, this notion was introduced in the context of p-adic number fields by Ore [19], but remained less well-known. In this article, we clarify and consolidate the results which can be reached concerning the factorization of I(x) by using these associated polynomials which we consider in the context of arbitrary local fields (Theorem l.5). The key idea consists in interpreting the associated polynomial as a residue class for a suitable valuation w of the field of rational functions E(X). This allows us to see the factorization of the associated polynomial as a factorization on the residue class field (with respect to the valuation w) of E(X) which can be lifted to a factorization of I(X) in E[X] with the help of a statement similar to that of Hensel's lemma. Moreover, we also obtain, as an important consequence of these considerations, a criterion for irreducibility expressed in terms of associated polynomials (Theorem 1.6). This criterion is easy to test and flexible enough to be applied in a great number of cases. Ore's theorem on the factorization of a prime number into prime ideals of a number field (Theorem 1.8) hence appears as a special case of our Theorem l.5, which is a genuine extension of Ore's theorem to a general local field. Moreover, our approach clarifies the causes of the validity of such results by giving a more modem and accessible proof. This makes the result applicable to many diverse situations, for instance to the so-called "generalized difference polynomials" . We obtain an irreducibility criterion (Proposition 5.1) over an arbitrary field K for the bivariate polynomials of the form n
I(T, X) =
L Pj(T)X"-jEK[T,
X].
(1)
j=O
Such a polynomial was defined by Abhyankar and Rubel [1] to be a generalized difference polynomial if PoEK, PoPn-:f:.O,and degPj(X),-j
(3) of f(X)
= n""Qt(X)
as
+ q>(X)Q(X),
j=O
with degree Qj < m. By hypothesis, modulo tt, we have, in .E[X],
the term n"'Qt(X)
J= n"'Qt(X)
is not zero. Reducing
+ cpQ.
It follows that n"'Qt(X) is the remainder in the Euclidean division of J by cp. Hence rp divides j in E[X] if and only if at;cO. Now at = means that the (E, q»-polygon of f(X) is reduced to a point or a horizontal side as required.
°
In the sequel of this section, we concentrate exclusively on monic polynomials with integer coefficients. We recall notations and definitions, mainly of Ore [19].
1 I
FACTORIZATION OVER LOCAL FIELDS
177
DEFINITION 1.3. Let f(X) E Adx] a monic polynomial with integer coefficients. The (E, q>)-polygon, minus its possible horizontal part, is called the principal part of it. We fix a monic polynomialf(X)EAE[X]. principal part of the (E, q»-polygon of f(X)
Let SI, ... , S» be the sides of the with increasing slopes. Denote by
10 = the length of the possible horizontal side; li= the length of the projection of Si to the x-axis; hi = the length of the projection of Si to the y-axis. Set
e, = (li, h;),
Ai
= l,/ e,
x, = hj e;
and
For a given side Si of the (E, q»-polygon of f(X),
consider the sum of the terms
najQj(X)q>(X)' - j to the points (j,
in (3) corresponding
aJ E Si.
In this sum, we separate
q>(xy-1o-" -linh,+ "+hi-" thus making apparent
a factor
K,o(X)q>(X)li + Ri,] (X)nK'iq>(X)li- "i + Ri,2(X)n2K'iq>(X)li- 2-'i + ... + Ri",,(X)n"', where the polynomials Ri,o(X), ... ,Ri,e,(X) are of degree less than m, with coefficients in AE. In particular K,o(X) and q>(X), considered as polynomials of E[X], are coprime. So there exists Ai(X)EAdX] such that
Ri,o(X)Ai(X)
== 1
mod (x, q>(X)).
Define
DEFINITION lA.
The associated polynomial
of the z-th side is
Fi(X, Y) = yei + Si, (X) ye,-] + ... + Si,ei(X)' Though the associated polynomial FJX, Y) depends class modulo the ideal (z, q>(X)) does not.
on the choice of Ai, its
Making use of these associated polynomials of sides, we are now able to state a general result about the factorization of polynomials over local fields. This is the main result of our paper. THEOREM 1.5.
Letf(X)EAE[X]
be a monic polynomial.
f(X) == q>](X)(l, ... q>s(xt' is the factorization of f(X) that it does not divide f(x).
Assume that
mod x,
modulo it, where each q>vCX)EAE[X] is chosen so Denote by m; the degree of q>v(X). Then
f(X) = ] (X) ... s(X) v where v(X)== q>vCX)amod tc.
in AE[X],
178
S. D. COHEN, A. MOVAHHEDI AND A. SALINIER
In order to obtain a further factorization of the polynomial (X) = v(X) corresponding to the irreducible factor cP CPvof f, construct the (E, cp)-polygon of f(X)· For each side Si of the principal part of this polygon, let Ai li/(li, hi) be the parameter defined above and consider the factorization modulo (x, cp) of the associated polynomial Fi(X, Y):
=
F(X I
Then (i)
,
Y)
= F(i) (X 1,
-
has a factorization
=
y)a)'J ... F(') (X l;'
in Adx]
Y)a~;J
mod (n, cp).
of the form k
/i
= n n T(X),
(X)
i= Ij= I
where Y\X) is of degree m mY) a)O Ai, with m)') = degy F)') (X, Y). If a is a root of w(P) + w(Q) - .uop. As, furthermore,
v(t It follows that proposition.
the inequality
(12) is strict, we have
Qk,/JO-k-I (S') + tRk,/JO-k(S'») w(PQ) ~ w(P) + w(Q),
= w(P)
which
+ w(Q) -
concludes
.uop.
the proof
(14) of the
The following lemma will be used in the proof of the so-called "theorem of the product" (Theorem 3.2).
182
S. D. COHEN, A. MOVAHHEDI
AND A. SALINIER
LEMMA 2.3. Let P and Q be two polynomials positive real number. Then
in E[X], and p be any strictly
and, similarly,
Proof We retain the notation introduced proposition. According to (14), we have v
(I.
Qk.~-k
-I
(s) +
k
I.
in the proof of the preceding
=
Rk.JlO-k(O) + J1.op w(PQ),
k
and, by (9), I(PQ) ~ I(P) + I(Q). then p w(PQ). Now, if k + I = p-1,
+ I(Q),
(15)
then by (12) we have v (Qk,,(S)) > w(PQ) -pp,
and if k + I = u, with P < I(P) + I(Q), necessarily k < I(P) or 1< I(Q) so that, by (11), v(Rk,t{ 0) > w(P) + w(Q) - (k + I)p = w(PQ) -pp, which shows the inequality (15). The equality S(PQ) = S(P) + S(Q) can be shown similarly. We will suppose, from here on, that p is a rational number K/A with K and A two natural coprime integers. By its definition, the valuation w corresponding to p takes its values in O/A)£:. We denote by tJ (resp. m) the valuation ring (resp. the valuation residue class in tJ /rn.
ideal) in E(X) of w. For PE C, we denote by Pits
PROPOSITION 2.4. The residue class X is a root of ifJ so that E(X) is isomorphic to E(~). The residue field tJ'/m is a purely transcendental extension of E(X) generated by the residue class Y of cp(X)"-/ it" modulo m. The residual image of any element of {1 of the form a(X)cp(xt, where deg (a) < m, is equal to zero if A does not divide k, and either 0 or a monomial in Y of degree k/A otherwise.
Proof Since cp(X) E m, the residual image of X is a root of lp. Therefore the field E(X) is isomorphic to E(~), where ~ is the residual image of S. The inclusion Adx] ~ {1 gives us, by reduction, a morphism of fields E(X)~C/m. We now show that the residue class Y of cp(X)A[n" modulo m is transcendental over E(X). Suppose that there exist polynomials ak(X)E Adx] of degree less
183
F ACTORIZA TION OVER LOCAL FIELDS
than m such that the polynomial Lkak(X)CP(XY,k /nKk belongs to the valuation ideal m, Then, by the definition of w, it follows that v(ak((» > 0, for all k. Hence, by Lemma 2.1, n divides all the coefficients of all the ak(X). This means that Uk = for all k, from which it follows that Y is transcendental over
°
ECK).
Next, we shall prove that every element of {l/m is a rational fraction in Y with coefficients in E(X). To this end, it suffices to justify the final statement of the proposition, For, by the definition of w, a polynomial of E[X] belongs to {l if and only if each term a(X)cp(Xt of its canonical decomposition belongs to (l. The w-valuation of a(X)cp(X)k is v(a(O) + k(K/ A). Since v(a(O) is a rational integer, the residue class of a(X)cp(Xt is zero, whenever A does not divide k. On the other hand, if A divides k and the residue class of a(X)cp(Xt is not zero, we have v(a(O) + k'K= 0, where k' = k/A, so that we can write a(X)cp(Xt with b(X)EAdx]
= n-k'Kb(X)cp(Xt'\
of degree less than m, and a(X)qJ(X)k = b(X) yk'.
Remark. Let Ea be the completion of the field Ea with respect to the valuation v. The valuation v is extended in a unique way to a continuous valuation of Ea, again denoted by v. There exists a valuation w of the field Ea(X) extending both v and w, It is known [16] that there are exactly three types of extensions of v to Ea (X). The Proposition 2.4 indicates that, whenever p is rational, w is an extension of the first type of v, so that Ea (X) is an "inert" extension of Ea. As explicit formulas are known [2, 16] for these extensions of the first type, it would be possible to define the valuation w corresponding to a rational number as the restriction to E(X) of such a valuation W. This alternative approach can be found in the literature ([2], [21, Theorem 1.2]). The following lemma links the degree of a polynomial in {l with the degree in Y of its residual image in (l/m, Recall that m denotes the degree of cp. LEMMA 2.5. (i) Let PE{lnE[X] and let 1'EE(X)[y] be its residue class modulo m, Then the degree of l' in Y and the degree of P in X satisfy the inequality
_) 0, w(T);3 0, w(S);3 O. Suppose the opposite of the desired conclusion, that is, w(VJb + Qa) > min (w(a), web)) for some (a, b) in A x B. Then necessarily we have w(a) = web). Let us put c = L(a, b) = VJb + Qa.
(20)
=
By Euclidean division of cS - aZ by VI, we can write cS - aZ V, Q' + R', with deg (R') < deg (V,). By Lemma 2.6, we have w(R') > w(a). Combining the two formulas (19) and (20), we get V, (Ta - Sb) = - Se + (l + Z)a, so that V, divides a - R'. Since the degrees of a and R' are both smaller than degx(V,), we conclude that a = R' and so w(a) < w(R') = w(a). This contradiction shows the desired equality w(V,b + Qa) = min (w(a), web)). Finally Lemma 2.7 yields the result.
§3. Relationship between Newton polygons and valuations. In this section, we fix a uniformising element n: of a local field E, and a monic polynomial O. The class F(Y) modulo (Jr, ep(X)) of the associated polynomial of the unique side of the (E, ep)-polygon of f(X) is, according to Proposition 3.4, obtained from f(X)/JrPI by reduction modulo m. A simple calculation gives F(Y) =
I
Qj(X)y(l-j)/J...
lXj=pj
As X and e are roots of the same irreducible polynomial ep(X)E E[X], and since F is known to be a power of an irreducible polynomial of.degree m' in E(X)[Y], we see that the polynomial I,lXj=pjQj(e)y(I-j)/J.. is also a power of an irreducible polynomial of degree m' in E( e)[Y]. On the other hand, replacing
FACTORIZATlO
X by
e in the canonical
191
OVER LOCAL FIELDS
decomposition of f(x), VCj~Pj Qj(e)/3(I-j)/A)
we get > O.
Hence, by reducing modulo the maximal ideal of the local field E(e), we derive
L Qj(e)/3(t-j)/A = 0, «i= pj
which shows that /3 is a root of an irreducible polynomial in E(e)[Y] of degree m'. We deduce that the degree of E(e, /3) over E is mm' and, consequently, mm' divides the residue degree of the local extension E(e)/ E.
§4. Proof of the main theorem. All the ingredients are in place for the proof of Theorem 1.5. Let f(X) E Ad X] be a monic polynomial. Assume that f(X) == is the factorization of f(X) f(X)
CPI
(xt' ... CPsCx)°'
mod
It,
modulo n, where CPy(X) E Adx]. = cI>1(X)
The factorization
... cI>s(X)
with cI>vCX)== cpvCX)Ovmod n, is a direct application of the classical version of Hensel's lemma. We fix one of these factors cI>(X) = cI>vCx) of f(x), corresponding to cp(X) = CPv(X), an irreducible factor of f(X) mod n, chosen so that it does not divide f(x). By the theorem of the product and Proposition 1.2, the principal part of the (E, cp)-polygon of f(X) is derived from the (E, cp)-polygon of cI>(X) by a horizontal translation, noting that the (E, cp)-polygon of cI>(X) has no horizontal side. On account of the hypothesis "cp(X) is not a factor of f(X)", the length of the horizontal projection of the (E, cp)-polygon of cI>(X) is a = ay. Let a
cI>(X) = cp(X)0 +
L
najQj(X)cp(X)0-j,
j=1
be the (n, cp)-canonical decomposition of cI>(X). We proceed by induction on the integer a. If a = 1, we see at once that k = 1 and also that I1 = AI = l. Thus the associated polynomial to this unique side is linear: tl = m\l) = a\l) = l. So (i) and (ii) hold immediately. Now assume that the result is true for integers less than a. We examine the factorization of the associated polynomial of the last side Sk: Fk(X, Y) == F\k)(X, y)a\k) ... F~;)(X, y)0\!)
mod (n, cp).
Let Gk(y) be the residue class of cI>(X)/niz,++iz modulo the maximal ideal m of the valuation w corresponding to the slope p; = 1(d Ak of Si, Reducing the coefficients modulo n and substituting X for the indetermi~ate X in the k
FACTORIZATION
193
OVER LOCAL FIELDS
ow Ore's theorem is an easy consequence of Theorem 1.5 by means of the well-known fact that the factorization of a polynomial f(X) E Q[X] over the local field Qp determines the decomposition of the prime number p in product of prime ideals in the number field Q[X]/(f). More precisely, we have the following proposition [18, Prop. 6.1]. PROPOSITION
into polynomials of the prime p:
4.1.
Let RK be the ring of integers of K. The factorization
irreducible in QAX] gives rise to the following
decomposition
where the Pu are distinct prime ideals of RK. If fu is the residue degree of Pu, then for each v the product eufu equals the degree of gu(X). Moreover, for V = 1,2, ... , s we have Kp, = Qp(8v) with gv(8v) = o.
Now the proof of Theorem 1.8 goes as follows. By Theorem 1.5 we have
where
