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Some extensions and applications

of Eisenstein Irreducibility

Criterion

Sudesh Kaur Khanduja

Department of Mathematics

Panjab University, Chandigarh

E-mail: skhand@pu.ac.in

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Eisenstein Irreducibility Criterion.(1850)

Let F(x) = a0xn+ a1xn−1+ ... + anbe a

polynomial with coefficient in the ring Z

of integers. Suppose that there exists a

prime number p such that

• a0is not divisible by p,

• aiis divisible by p for 1 ≤ i ≤ n,

• anis not divisible by p2

F(x) is irreducible over the field Q of ra-

tional numbers.

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Example: Consider the pth cyclotomic

polynomial xp−1+ xp−2+ ... + x + 1 =

xp−1/x − 1 On changing x to x+1 it be-

comes (x + 1)p− 1/(x + 1 − 1) = xp−1+

?p

irreducible over Q.

1

?

xp−2+ ... +

?

p

p − 1

?

and hence is

This slick proof of the irreducibility for the

pth cyclotomic polynomial was given by the

Eisenstein,thoughitsirreducibiltywasproved

by Gauss in 1799.

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In 1906, Dumas proved the following gen-

eralization of this criterion.

Dumas Criterion. Let F(x) = a0xn+

a1xn−1+ ... + an be a polynomial with

coefficients in Z. Suppose there exists a

prime p whose exact power pridividing

ai(where ri= ∞ if ai= 0), 0 ≤ i ≤ n,

satisfy

• r0= 0,

• (ri/i) > (rn/n) for 1 ≤ i ≤ n − 1 and

• gcd(rn, n) equals 1.

Then F(x) is irreducible over Q.

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Example : x3+3x2+9x+9 is irreducible

over Q.

Note that Eisenstein’s criterion is a special

case of Dumas Criterion with rn= 1.

For a given prime number p, let vpstand

for the mapping vp : Q∗→ Z defined as

follows. Write any non zero rational number

x = pra

b,p ? ab. Set vp(x) = r. Then

(i) vp(xy) = vp(x) + vp(y)

(ii) vp(x + y) ≥ min{vp(x),vp(y)}.

Set vp(0) = ∞. vpis called the p-adic val-

uation of Q.

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Dumas Criterion. Let F(x) = a0xn+

a1xn−1+ ... + an be a polynomial with

coefficients in Z. Suppose there exists a

prime number p such that vp(a0) = 0,

vp(ai)/i > vp(an)/n for 1 ≤ i ≤ n − 1

and vp(an) is coprime to n, then F(x) is

irreducible over Q.

In 1923, Dumas criterion was extended to

polynomials over more general fields namely,

fields with discrete valuations by K¨ ursch´ ak.

Indeed it was the Hungarian Mathematician

JOSEPH K¨URSCH´AK who formulated the

formal definition of the notion of valuation

of a field in 1912.

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Definition :A real valuation v of a field K

is a mapping v : K∗→ R satisfying

(i) v(xy) = v(x) + v(y)

(ii) v(x + y) ≥ min{v(x),v(y)}

(iii) v(0) = ∞.

v(K∗) is called the value group of v. A

real valuation is said to be discrete if v(K∗)

is isomorphic to Z.

In 1931, Krull generalized the above notion

of valuation.

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By a Krull valuation of a field K we mean

a mapping

v : K∗→ G

where G is a totally ordered (additively writ-

ten) abelian group satisfying (i), (ii) and

(iii). The pair (K,v) is called a valued field.

The subring Rv= {x ∈ K| v(x) ? 0} of

K is called the valuation ring of v. It has a

unique maximal ideal given by Mv= {x ∈

K| v(x) > 0}. Rv/Mvis called the residue

field of v. For any ξ belonging to Rv¯ξ will

stand for the canonical homomorphism from

Rvonto Rv/Mv.

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Example. Let vxdenote the x-adic val-

uation of the field Q(x) of rational functions

in x trivial on Q and vpdenote the p-adic

valuation of Q. For any non-zero polyno-

mial f(x) belonging to Q(x), we shall de-

note by f∗the constant term of the polyno-

mial f(x)/xvx(f(x). Let v be the mapping

from non-zero elements of Q(x) to Z × Z

(lexicographically ordered) defined on Q[x]

by

v(f(x)) = (vx(f(x)),vp(f∗)).

Then v gives a valuation on Q(x).

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In 1997, Saha and S.K.K generalized the

Eisenstein-Dumas-K¨ ursch´ ak criterion.

Theorem1.(-,Saha) let v be a Krull val-

uation of a field K with value group G

and F(x) = a0xs+ a1xs−1+ ... + asbe a

polynomial over K. If

• v(a0) = 0,

• v(ai)/i ≥ v(as)/s for 1 ≤ i ≤ s and

• there does not exit any integer d > 1

dividing s such that v(as)/d ∈ G,

then F(x) is irreducible over K.

Definition. A polynomial which satisfies

the hypothesis of Theorem 1 is called an

Eisenstein-Dumas polynomial with respect

to v.

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Example: Let F(X,Y ) = g(Y )Xs+h(Y )

be a polynomial over a field L in indepen-

dent variables X,Y . If g(Y ),h(Y ) have no

common factors and deg g(Y ) − deg h(Y )

is coprime to s,then F(X,Y ) is irreducible

over L.

Verification: Regard F(X,Y )/g(Y ) as

a polynomial in X with coefficients over the

field K = L(Y ) with valuation on K defined

by v(a(Y )/b(Y )) = deg b(Y ) − deg a(Y )

and apply the criterion by Saha.

In 2001, S. Bhatia generalized Eisenstein’s

Irreducibility Criterion in a different direc-

tion.

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Theorem 2. (–, S. Bhatia) Let v be a val-

uation of a field K with value group the

set of integers. Let g(x) = xm+a1xm−1+

... + ambe a polynomial with coefficients

in K such that v(ai)/i > v(am)/m for

1 ≤ i ≤ m−1. Let r denote gcd(v(am),m)

and b be an element of K with v(b) =

v(am)/r. Suppose that the polynomial zr+

(am/br)−in the indeterminate z is irre-

ducible over the residue field of v. Then

g(x) is irreducible over K.

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Theorem 3. Let f(x) and g(y) be non-

constant polynomials with coefficients in

a field k. Let c and c0denote respectively

the leading coefficients of f(x) and g(y)

and n, m their degrees. If gcd(m,n) = r

and if zr− (c0/c) is irreducible over k,

then so is f(x) − g(y).

The result of Theorem 3 has its roots in

a theorem of Ehrenfeucht. In 1956, Ehren-

feucht proved that a polynomial f1(x1) +

.... + fn(xn) with complex coefficients is ir-

reducible provided the degrees of f1(x1),...,

fn(xn) have greatest common divisor one.

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In 1964, Tverberg extended this result by

showing that when n ≥ 3, then f1(x1) +

.... + fn(xn) belonging to K[x1,....,xn] is

irreducible over any field K of characteristic

zero in case the degree of each fiis positive.

Of course this result is false when character-

istic of K is p > 0. Note that if a polyno-

mial F can be written as F = (g1(x1))p+

(g2(x2))p+ ..... + (gn(xn))p+ c[g1(x1) +

g2(x2) + .... + gn(xn)] where c is in K and

each gi(xi) is in K[xi], then it is reducible

over K.

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In 1966, Tverberg proved that the converse

of the above simple fact holds in the partic-

ular case when n = 3 and K is an alge-

braically closed field of characteristic p > 0.

In 1982, Schinzel extended Tverberg’s result

by showing that this converse holds for any

n ≥ 3. In 2004, Amrit Pal has given a proof

of Schinzel’s result which is shorter and en-

tirely different from Schinzel’s proof.

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Question: When is a translate g(x+a) of

a given polynomial g(x) with coefficients

in a valued field (K,v) an Eisenstein-Dumas

polynomial with respect to v?

In 2009, we have characterized such poly-

nomials using distinguished pairs.

Theorem 4 (-, Anuj Bishnoi). Let v be

a henselian Krull valuation of a field K.

Let g(x) belonging to Rv[x] be a monic

polynomial of degree e having a root θ.

Then for an element a of K, g(x + a)

is an Eisenstein-Dumas polynomial with

respect to v if and only if (θ,a) is a dis-

tinguished pair and K(θ)/K is a totally

ramified extension of degree e.

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The following result which generalizes a re-

sult of M. Juras [12] proved in 2006 has been

quickly deduced from the above theorem.

Theorem 5. Let g(x) =

e

?

i=0

aixibe a

monic polynomial with coefficients in a

henselian valued field (K,v). Suppose that

the characteristic of the residue field of v

does not divide e. If there exists an ele-

ment b belonging to K such that g(x + b)

is an Eisenstein-Dumas polynomial with

respect to v, then so is g(x −ae−1

e).

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Classical Sch¨ onemann Irreducibility

Criterion.(1846) If a polynomial F(x)

belonging to Z[x] has the form F(x) =

φ(x)s+pM(x) where p is a prime number,

• φ(x) belonging to Z[x] is a monic poly-

nomial which is irreducible modulo p,

• φ(x) is co-prime to M(x) modulo p,

and

• the degree of M(x) is less than the de-

gree of F(x),

then F(x) is irreducible in Q[x].

Eisenstein’s Criterion is easily seen to be a

particular case of Sch¨ onemann Criterion by

setting φ(x) = x.

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In 1997, Saha gave a generalization of Clas-

sicalSch¨ onemannIrreducibilityCriterionus-

ing the theory of prolongations of a valua-

tion defined on K to a simple transcenden-

tal extension of K which was initiated by

MacLane and developed further by Popescu

at el. In 2008, Ron Brown has given a dif-

ferent proof of Saha’s result.

Recently, we have extended the General-

izedSch¨ onemann-EisensteinIrreducibilityCri-

terion.

Theorem 6. (-, R. Khassa) Let v be a dis-

crete valuation of K with value group Z

and π be an element of K with v(π) = 1.

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Let f(x) belonging to Rv[x] be a monic

polynomial of degree m such that¯f(x) is

irreducible over Rv/Mv.

longing to Rv[x] be a monic polynomial

Let F(x) be-

having f(x)-expansion

n

?

i=0

Ai(x)f(x)i. As-

sume that there exists s ? n such that

π does not divide the content of As(x),

π divides the content of each Ai(x),0 ?

i ? s − 1 and π2does not divide the

content of A0(x).Then F(x) has an irre-

ducible factor of degree sm over the com-

pletion (ˆK, ˆ v) of (K,v) which is a

Sch¨ onemann polynomial with respect to ˆ v

and f(x).

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Theorem 7. Let (K,v),π be as above

and F(x) = xn+an−1xn−1+...+a0be a

polynomial over Rvsatisfying the follow-

ing conditions for an index s ? n − 1.

(i) π|aifor 0 ? i ? s − 1, π2? a0,π ? as.

(ii) The polynomial xn−s+¯ an−1xn−s−1+

...+¯ asis irreducible over the residue field

of v.

(iii)¯d ?= ¯ asfor any divisor d of a0in Rv.

Then F(x) is irreducible over K.

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Gotthold Max Eisenstein

Born: 16 April 1823 in Berlin, Germany

Died: 11 Oct 1852 in Berlin, Germany

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What attracted me so strongly and ex-

clusively to mathematics, apart from the

actual content, was particularly the spe-

cific nature of the mental processes by which

mathematical concepts are handled. This

way of deducing and discovering new truths

from old ones, and the extraordinary clar-

ity and self-evidence of the theorems, the

ingeniousness of the ideas ... had an ir-

resistible fascination for me. Beginning

from the individual theorems, I grew ac-

customed to delve more deeply into their

relationships and to grasp whole theories

as a single entity. That is how I conceived

the idea of mathematical beauty ...

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PotenzenvonPrimzahlensind, Journal f¨ ur die Reine

und Angew. Math. 32 (1846) 93-105.

2. G. Eisenstein,¨Uber die Irreduzibilitat und einige

andere Eigenschaften der Gleichungen, Journal f¨ u

die Reine und Angew. Math., 39 (1850) 160-179.

3. G. Dumas, Sur quelques cas d’irreductibilite des

polynomes` acoefficientsrationnels, Journal de Math.

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4. J. K¨ ursch´ ak, Irreduzible Formen, Journal f¨ u die

Reine und Angew. Math., 152 (1923) 180-191.

5. S. Maclane, The Sch¨ onemann - Eisenstein irre-

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6. A. Ehrenfeucht, Kryterium absolutnej nierozk-

ladalnosci wielomianow, Prace Math., 2 (1956) 167-

169.

Page 25

7. H. Tverberg, A remark on Ehrenfeucht’s crite-

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Page 26

13. S. Bhatia and S. K. Khanduja, Difference poly-

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