EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 2, 2024, 721-724
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Note on irreducible polynomials over Fq[X]

Alanod M. Sibih

Department of Mathematics, Jamoum University College, Umm Al-Qura University,
Holly Makkah 21955, Saudi Arabia

Abstract. In this note, we provide a new criterion of polynomials’s irreducibility over Fq[X],
where Fq is a finite field.

2020 Mathematics Subject Classifications: 11Txx, 11T55

Key Words and Phrases: Polynomials, irreducibility, criterion, finite fields.

1. Introduction

A polynomial is reducible over a given field if it can be expressed as a product of
lower degree polynomials with coefficients in the same field. Otherwise, it is called to be
irreducible.

We are interested in determining if a particular polynomial is irreducible or not. As a
result, a simple test or criterion for obtaining this information is desirable.

Unfortunately, no such criterion that applies to all classes of polynomials has yet been
developed; nonetheless, a number of tests, or irreducibility criteria, have been discovered
so far that provide useful information for some specific classes of polynomials.

This article focuses on irreducible polynomials with coefficients in Fq[X], where over
Fq is a finite field.

A. Chandoul et al. [2], proved a widely accepted irreducibility criterion, which states
that:

Theorem 1. If Λ(Y ) = Y d+λd−1Y
d−1+ · · ·+λ0 be a polynomial with λi ∈q [X], λ0 ̸= 0

and deg λd−1 > deg λi, for each i ̸= d− 1. Then Λ is irreducible over q[X].

This result was the starting point for many researches and the exploration of new
criterions, see [1, 3]. For older results, see [4, 5]. In this note, we provide a new criterion
of polynomials’s irreducibility over Fq[X].

DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5095

Email address: amsibih@uqu.edu.sa (A. M. Sibih)

https://www.ejpam.com 721 © 2024 EJPAM All rights reserved.



A. M. Sibih / Eur. J. Pure Appl. Math, 17 (2) (2024), 721-724 722

2. Preliminaries

Let Fq be the finite field and denote by Fq[X] the ring of polynomials with coefficients
in Fq and by Fq(X) the quotient field of Fq[X]. Let Fq((X

−1)) be the field of Laurent
formal power series defined as follows:

Fq((X
−1)) = {

∑
n≥n0

anX
−n, an ∈ Fq and n0 ∈}.

For w =
+∞∑
n=n0

anX
−n ∈ Fq((X

−1)), we define the integer part [w] of w by [w] =
0∑

n=n0

anX
−n

if n0 ≤ 0 and [w] = 0 if n0 > 0, the fractional part of w by {w} = w− [w] =
+∞∑

n = 1

anX
−n.

We have a non-archimedean absolute value | · | on Fq((X
−1)), namely, for any element

w ∈ Fq((X
−1)) having the form

w =

+∞∑
n=n0

anX
−n (an ∈ Fq),

we define |w| = e−n0 if w ̸= 0, where n0 is the smallest index verifying an0 ̸= 0, and
|w| = 0 if w = 0. We know that Fq((X

−1)) is complete and locally compact with respect
to the metric defined by this absolute value.

We denote by Fq((X
−1)) an algebraic closure of Fq((X

−1)). We note that the absolute
value has a unique extension to Fq((X

−1)). To denote this extended absolute value, we
also use the symbol | · |.

3. Main results

Theorem 2. Let Fq be a finite field of caracteristic p, n ≥ 2 and let

P (Y ) = AsY
s +As−1Y

s−1 +As−2Y
s−2 + · · ·+A1Y +A0

be a polynomial over Fq[X], such that AsAs−1A0 ̸= 0, As and As−1 has a same irreducible
factor B, with lcm(As−1, B) = Bm (As−1 = Bmas−1) and lcm(As, B) = Bn (As = Bnas).
If

n > ms+
(s− 1)(degAs −mdegB) +M

degB

with M = max(deg
i ̸=s

Ai), then P is irreducible over Fq[X].

Proof. Suppose that P (Y ) = Q(Y )H(Y ), where Q,H ∈ Fq[X][Y ]. let



A. M. Sibih / Eur. J. Pure Appl. Math, 17 (2) (2024), 721-724 723

Q(Y)= QjY
j +Qj−1Y

j−1 +Qj−2Y
j−2 + · · ·+Q1Y +Q0

and H(Y)= HkY
k +Hk−1Y

k−1 +Hk−2Y
k−2 + · · ·+H1Y +H0

where j + k = s, QjHk = As, Q0H0 = A0 and As−1 = QjHk−1 + HkQj−1. Let
Bd = lcm(Qj , B), (Qj = Bdqj), then Bn−d = lcm(Hk, B) (Hk = Bm−dhk) and we must
have m ≥ d.

Consider the factorisation of P and Q in Fq((X−1)), we have

P(Y)=As(Y − ω1) · · · (Y − ωn)
and Q(Y)=Qj(Y − ω1) · · · (Y − ωj)

where ωi ∈ Fq((X−1)), forall i := 1, · · · , n.
Consider, now, the nonarchimedean absolute value, and set a real number α ≥ 0 such

that
|As| > eαmax |Ai|

i ̸=s

then, using the viète theorem, we have

|ω1 · · ·ωs| = |ω1| · · · |ωs| =
|A0|
|As|

<
|A0|

eαmax |Ai|
i ̸=s

<
1

eα
,

thus, for any j := 1, · · · , n, we must have |ωj | <
1

eα/s
.

So that, we get

|ω1 · · ·ωj | <
1

ejα/s
.

On the other hand, we have

|ω1 · · ·ωj | =
∣∣∣∣Q0

Qj

∣∣∣∣ = ∣∣∣∣ Q0

Bdqj

∣∣∣∣ ≥ 1

|Bm| |as|
.

To reach a contraduction, it is still necessary to chose α such that

1

|Bm| |as|
≥ 1

ejα/s
.

It can be sufficient to choose α such that

|Bm| |as| ≤ eα/s.

Or, equivalently

α ≥ smdegB + s(degAs − n degB).

A conceivable value for α is sm degB + s(degAs − n degB), which leads to a contra-

diction if n > ms+
(s− 1)(degAs −m degB) +M

degB
where M = max(deg

i ̸=s
Ai), what was to

be proved.



REFERENCES 724

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