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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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