Irreducibility of Polynomials with Square Coefficients over Finite Fields

Abstract

We study a random polynomial of degree n over the finite field Fq\mathbb{F}_q, where the coefficients are independent and identically distributed and uniformly chosen from the squares in Fq\mathbb{F}_q. Our main result demonstrates that the likelihood of such a polynomial being irreducible approaches 1/n+O(q−1/2)1/n + O(q^{-1/2}) as the field size q grows infinitely large. The analysis we employ also applies to polynomials with coefficients selected from other specific sets.

Full text

arXiv:2410.16814v1 [math.NT] 22 Oct 2024
IRREDUCIBILITY OF POLYNOMIALS WITH SQUARE COEFFICIENTS
OVER FINITE FIELDS
LIOR BARY-SOROKER AND ROY SHMUELI
Abstract. We study a random polynomial of degree nover the finite field Fq, where the
coefficients are independent and identically distributed and uniformly chosen from the squares
in Fq. Our main result demonstrates that the likelihood of such a polynomial being irreducible
approaches 1/n +O(q−1/2) as the field size qgrows infinitely large. The analysis we employ
also applies to polynomials with coefficients selected from other specific sets.
1. Introduction
Consider the random polynomial
(1) Pξ(x) = xn+ξ1xn−1+ξ2xn−2+···+ξn,
where the coefficients ξ1, . . . , ξnare independent random variables taking values in a ring R. The
main question of interest is what is the probability that Pξ(x) is irreducible over R.
When R=Z, one expects the probability to be close to 1. To clarify this, we focus on two
central models. The first, often called the large box model, assumes that the degree nis fixed and
the coefficients ξiare uniformly distributed within the interval {−H,...,H}as H→ ∞. In this
case, not only is Pξ(x) asymptotically almost surely (a.a.s.) irreducible, but the Galois group is
also the full symmetric group a.a.s. The main challenge in this model is to bound the error term
and demonstrate that it arises from reducible polynomials (see [vdW36,Kno57,Che63,Gal73,
Kub09,Zyw10,Die13,CD20,AGO+21,CD23a,CD23b]). This problem remains open despite a
recent major breakthrough [Bha21].
In the second model, while the coefficients do not grow, the degree does—for example, when
ξiare Rademacher variables. Here, we always condition on ξn6= 0. In this context, [BV19]
proved that under the General Riemann Hypothesis (GRH), the polynomial is irreducible a.a.s.
Unconditional results [BSK20,BSKK23] show that the probability is bounded away from zero
in full generality and under mild restrictions tend to 1 (e.g., ξiare uniformly distributed on an
interval of length ≥35). See [BSG24] for a treatment of a hybrid model in which ξiare as in the
large box model, but nmay grow arbitrarily fast as a function of H.
In this work we assume that Ris a finite field. Let qbe a prime power and R=Fqthe finite
field with qelements. In this setting, the ring of polynomials Fq[x] is closely analogous to the
ring of integers Z, and therefore the irreducible polynomials are analogous to prime numbers.
And indeed, the first result is the prime polynomial theorem [Ros02, Theorem 2.2] which may be
interpreted as the statement that if the ξiare uniform on Fq, then
(2) PhPξ(x) irreducible over Fqi=1
n+Oq−n/2,
as qn→ ∞. There are variants when we fix some of the coefficients, see [Pol13,Ha16]. Moti-
vated by Maynard’s theorem [May19] on primes without the digit 7, Porritt [Por19] considered
polynomials with coefficients that are uniformly distributed in a subset S( Fq. He proved that if
#S≥q−√q/2 then
PhPξ(x) irreducible over Fqi∼q
n(q−1),
as n→ ∞. See [Mos17] for generalizations.
1

2 LIOR BARY-SOROKER AND ROY SHMUELI
In the case that nis fixed, there are much stronger results. In [Coh72,BBSR15], one can fix
all coefficients but three, under very mild assumptions. A general criterion by Entin [Ent21] deals
with uniform coefficients in a subset S⊆Fn
qthat is “regular” in some precise sense:
(3) PhPξ(x) irreducible over Fqi=1
n+Onq−1/2·irreg(S),
where irreg(S) is an explicit constant that depends on S, see (15). This asymptotic formula is
useful in several classical settings, such as when Sis a product of arithmetic progressions.
However, when qis odd and S=α2α∈Fqnis the set of vector with square entries, the
error term in (3) is bigger than the main term: irreg(S)≥q1/2−1n, see (16). Thus (3) gives
no non-trivial information. Our main result says that the probability for irreducibility is still
asymptotically 1/n:
Theorem 1. Let Pξ∈Fq[x]be a random polynomial as in (1)with ξ1,...,ξnuniformly distributed
in α2α∈Fq⊆Fq. Then
PhPξ(x)irreducible over Fqi=1
n+Onq−1/2,
as q→ ∞.
If qis even, then ξiare uniform in Fq, hence it follows from (2). So from now on we assume q
is odd.
Theorem 1 follows from a more general result, in which the coefficients distribute uniformly in
subsets of the following form. For M > 0, we say that a subset U⊆Fqis of complexity ≤Mif
there exist
f11,...f1k1,...,fm1,...,fmkm∈Fq[x]
of positive degree such that
(4) U=
m
[
i=1
ki
\
j=1
fij (Fq) and max
i=1,...,m
j=1,...,ki
{m, ki,deg fij } ≤ M.
For example, the set of squares is of complexity ≤2. Note that every subset Uof Fqis of
complexity ≤qsince
U=[
α∈U
fα(Fq) where fα(x) = xq−x+α.
Theorem 2. Let qbe an odd prime power, and U1,...,Un⊆Fqnonempty subsets of complexity
≤M. Let Pξ∈Fq[x]be a random polynomial as in (1)with ξ1,...,ξnuniformly distributed in
U1,...,Unrespectively. Then
PhPξ(x)irreducible over Fqi=1
n+On,M qn−1/2·#U−1
1···#U−1
n,
as q→ ∞.
The error term goes to zero if #Ui≫Mqfor all i(in the sense, #Ui≥CMq). This happens in
many cases, for example, if Ui=fi(Fq), for 1 ≤deg fi≤M. In some cases, e.g. U={α2}∩{−α2}
and q≡3 mod 4 we have #U≪M1. We discuss this further in section 6.
Acknowledgements. The authors thank Alexei Entin for helpful discussions.
This research was supported by the Israel Science Foundation (grant no. 366/23).
2. Preliminaries
This section establishes the notation and revisits foundational facts necessary for the principal
results.

IRREDUCIBILITY OF POLYNOMIALS WITH SQUARE COEFFICIENTS OVER FINITE FIELDS 3
2.1. Splitting types. Let Sndenote the set of n-tuples s= (s1,...,sn) of non-negative integers
si≥0 with the constraint s1+ 2s2+···+nsn=n. The splitting type of a permutation σ
in the symmetric group Sym(n) on nletters is defined as SplitType(σ) := s∈ Snsuch that si
represents the number of cycles of length iin the disjoint cycle decomposition of σ. Therefore,
two permutations in Sym(n) are conjugate if and only if they share the same splitting type. For
s∈ Sn, we denote by Csthe conjugacy class in Sym(n) comprising all permutations with splitting
type s. The size of Csis determined by Cauchy’s formula:
(5) #Cs=n! n
Y
i=1
isi·si!!−1
.
Let kbe a field, and let P∈k[x] be a polynomial of degree n. The splitting type of Pis
SplitType(P) := s∈ Snwhere siis the number of irreducible factors of Pof degree i, counted
with multiplicity.
2.2. Discriminant. Let kbe a field and let P∈k[x] be a separable polynomial. Denote by kP
the splitting field of Pover kand set G= Gal(kP/k). Let z1,...,znbe the roots of Pin kP, and
identify Gwith a subgroup of Sym(n) by the action of Gon the roots of P.
The discriminant of Pis defined as
DiscxP=Y
1≤i<j≤n
(zi−zj)2∈k×.
For σ∈G, we have that σ√DiscxP= sgn(σ)√DiscxPwhere sgn(σ) is the sign of the permu-
tation σ. Thus, the following statement is immediate from the fundamental theorem of Galois
theory:
Proposition 3. Let kbe a field of characteristic 6= 2. Then G6≤ Alt(n)if and only if DiscxPis
not a square in k.
2.3. Chebotarev’s Density Theorem. We briefly introduce an explicit and uniform version of
Chebotarev’s Density Theorem over finite fields. All proofs can be found in [ABSR15, Appendix
A] in the language of rings or in [Ent19,§4] in the language of schemes.
Let Fbe a field which is a regular extension of the finite field Fq. Let OFbe an integrally
closed finitely generated Fq-algebra with fraction field F. Let K/F be a finite Galois extension
with Galois group G, let OKbe the integral closure of OFin K, and let Fqrbe the relative
algebraic closure of Fqin K. In particular, Fqris contained in OK. For each v≥0, we define
Gv=nσ∈Gσ(α) = αqvfor all α∈Fqro.
Then G0≤Gis the kernel of the restriction map G→Fqr. If v≥1, then Gvis a set on which
G0acts by conjugation. Let C(Gv) denote the set of orbits.
Remark 4.If Kis regular over Fq, then r= 1, hence Gv=Gfor all v≥0.
Let Φ ∈HomFqrOK,¯
Fqbe unramified over OF(in the sense that ker Φ is unramified). Then,
there exists a unique element in G, which we call the Frobenius element, and we denote by
OK/OF
Φ∈G,
such that for all u∈ OF
(6) ΦOK/OF
Φu= Φ(u)q.
If we fix a surjective φ∈HomFq(OF,Fqv) that is unramified in OK, then the set
OK/OF
φ=OK/OF
ΦΦ∈HomFqOK,¯
Fqextends φ⊆Gv,

4 LIOR BARY-SOROKER AND ROY SHMUELI
is invariant under conjugation from G0, since hOK/OF
Φiτ=hOK/OF
Φ◦τi,τ∈G0. Moreover, G0acts
transitively on OK/OF
φ, so OK/OF
φ∈ C(Gv).
We use a complexity notion cmp(OF) = cmp(OF/OF) and cmp(OK/OF) as defined in [Ent19,
Section 4] in the language of schemes. In particular, the complexity of
Fqx1,...,xn, f −1
0/(f1(x1,...,xn),...,fm(x1,...,xn)),
is bounded by max(n, deg f0,...,deg fm), and the complexity of S/R is a bounded by a function
of cmp(R), cmp(S), and the degree of the extension S/R.
By the Lang-Weil’s estimates [LW54], the number of ramified φis Ocmp(OK/OF)(qn−1). We
shall use this bound in the rest of the work multiple times.
In this language, Chebotarev’s theorem gives a quantitative equidsitribution of OK/OF
φin
C(Gv): For C∈ C(Gv), we define
δCheb,v(C) = 1
qvd ·#(φ∈HomFq(OF,Fqv)
φsurjective, unramified in OK
and OK/OF
φ=C).
By the Lang-Weil estimates [LW54], the number of surjective φ∈HomFq(OF,Fqv) is equal to
qvd +O(qv(d−1)), so δCheb,v(C) is the asymptotic density of such φ-s with OK/OF
φ=C.
Theorem 5 (Chebotarev’s Density Theorem).Let v≥1and C∈ C(Gv). Then
(7) δCheb,v(C) = #C
#Gv
+Ocmp(OK/OF)q−1/2.
Proof. See [ABSR15, Theorem A.4]1and [Ent19, Theorem 3]. 
Remark 6.In [ABSR15] the authors require that OK/OFis unramified. This requirement may
be added by localizing at the ramified locus, and by the Lang-Weil estimates [LW54] applied to
the discriminant of OK/OF. This will change the formula by Ocmp(OK/OF)q−1, hence will not
affect (7).
Remark 7.Obviously, Theorem 5 generalizes to any G0-invariant subset of C⊆Gv. Indeed, such
Cis the disjoint union of elements in C(Gv). Note if C=∅, then δCheb,v (C) = 0.
We state and prove a well-known corollary that will be used in the sequel.
Corollary 8. In the notation above, assume that K=LM for L, M linearly disjoint Galois
extensions of F, and Lregular over Fq. Let A= Gal(L/F ),B= Gal(M/F ), let v≥0, let C⊆A
be a conjugacy class, and D∈ C(Bv). Then, C×D∈ C(Gv)and
δCheb,v(C×D) = δCheb,1(C)·δCheb,v(D) + Ocmp(OK/OF)q−1/2.
Proof. We may identify G∼
=A×Bvia the map σ7→ (σ|L, σ|M). Since Lis regular over Fqand
since L, M are linearly disjoint over F, it follows that the algebraic closures of Fqin Mand in K
coincide. Hence, Gv=A×Bvand C×D∈ C(Gv). Finally, if φ∈HomFq(F, Fqv) is surjective
and unramified in K, then by (6) one may readily deduce that
OK/OF
φ=OL/OF
φ,OM/OF
φ.
The assertion follows from Theorem 5 applied to each of the terms, and noting that cmp(OL/OF),
cmp(OM/OF) are bounded in terms of cmp(OK/OF) and that the number of φ∈HomFq(F, Fqv)
that are ramified in any of the fields K, L, M is Ocmp(OK/OF)q−1.
1The implied constant in loc.cit. is written to depend only on cmp(OF) and [K:F]. This is not true, and the
correct error term is as written here.

IRREDUCIBILITY OF POLYNOMIALS WITH SQUARE COEFFICIENTS OVER FINITE FIELDS 5
2.4. Two examples. We explicitly compute two examples that will be used in the proof of the
main result.
Example 9. Let t= (t1,...,tn) be an n-tuple of indeterminates. The generic polynomial of
degree nis defined to be
P(t;x) = xn+t1xn−1+···+tn∈Fq(t)[x].
Let F=Fq(t), let FPbe the splitting field of Pover F. Then G= Gal(FP/F )∼
=Sym(n)
and FP/Fqis regular. Moreover, φ∈HomFq(F, Fq) is unramified in FPif and only if Pφ:=
P(φ(t); x)∈Fq[x] is separable.
From (6) it follows that if φ∈HomFq(F, Fq) is unramified in FP, then
SplitTypePφ= SplitTypeOFP/OF
φ.
Since cmp(OFP/OF) is bounded by a function in nand since the number of ramified φis On(q−1),
Theorem 5 and (5) imply that for any s∈ Snwe have
1
qnφ∈HomFq(F, Fq)SplitTypePφ=s
=1
qnφ∈HomFq(F, Fq)φramified or SplitTypeOFP/OF
φ=s
= n
Y
i=1
isi·si!!−1
+On(q−1/2).
(8)
Example 10. Consider univariate polynomials fij (x)∈Fq[x] of degree >0, for i= 1,...,n and
j= 1,...,ki. Define
Si=
ki
\
j=1
fij (Fq)⊆Fq.
We may assume without loss of generality that fij(x)−tiis separable in x. If this is not the case,
fij (x) = ˜
fij (xpr) for some r > 0 and ˜
fij (x)−tiseparable. Moreover, fij (Fq) = ˜
fij (Fq), so we may
replace fij by ˜
fij without altering the sets Si.
Let Kij be the extension of Fq(ti) generated by a root of fij (x)−ti, let Nij the Galois closure of
Kij over Fq(ti), Ni=Qki
j=1 Nij , and N=N1···Nn. Then N/F is Galois. Let GN= Gal(N/F ),
Gij = Gal(Nij /Fq(ti)), Hij = Gal(Nij/Kij ), and πi,j :GN→Gij the restriction map.
Let φ∈HomFq(OF,Fq) be unramified in ON. Then, φi=φ|Fq(ti)is unramified in Nij for
all i, j. We have that φ(ti) = φi(ti)∈fij (Fq) if and only if φiextends to Φij :ONij →¯
Fqwith
Φij (OKij ) = Fqif and only if φiextends to Φij :ONij →¯
Fqsuch that hONij /Fq[ti]
Φij i∈Hij if and
only if OK/OF
φi∩Hij 6=∅. In other words, φ(ti)∈fij (Fq) if and only if ON/OF
φ∩π−1
ij (Hij )6=∅.
Therefore,
φ(t)∈S1× · ·· × Sn⇐⇒ ON/OF
φ∩π−1
ij (Hij )6=∅,∀i, j.
Now by Theorem 5 applied to the GN,0-invariant set
Ω = {σ∈GN,1:∀i, j, ∃σ0∈GN,0such that σ0σσ−1
0∈π−1
ij (Hij )}
we get that
(9) δCheb,1(Ω) = #Ω
#GN,1
+OM(q−1/2),

6 LIOR BARY-SOROKER AND ROY SHMUELI
where M= maxi,j {deg fij }and cmp(ON/OF) is bounded in terms of M. We finish this example
by adding the ramified homomorphisms.

1
qn
n
Y
i=1
#Si−δCheb,1(Ω)≤1
qn#φ∈HomFq(OF,Fq)φramified in Nand
φ(t)∈S1× · ·· × Sn
=OM(q−1).
(10)
3. General Theorem
Our most general theorem is stated below. We begin by showing how Theorem 1 and Theorem 2
follow from it. The remainder of the paper focuses on proving this theorem.
Theorem 11. Let qbe an odd prime power, s∈ Sn, and let U1,...,Un⊆Fqbe subsets of
complexity ≤M. Let Pξ∈Fq[x]be a random polynomial as in (1)where ξ1,...,ξnare uniformly
distributed in U1,...,Unrespectively. Then
PhSplitTypePξ=si= n
Y
i=1
isi·si!!−1
+On,M qn−1/2·#U−1
1···#U−1
n,
as q→ ∞.
Proof of Theorem 2.As Pξis irreducible if and only if SplitTypePξ= (0,...,0,1), and in this
case (Qn
i=1 isi·si!)−1= 1/n, we see that Theorem 2 is a special case of Theorem 11.
Proof of Theorem 1.As mentioned above, if qis even, then the coefficients are uniform, hence
we are done by (2). If qis odd, then Theorem 1 follows from Theorem 2 applied to the sets
U1=···=Un=α2α∈Fqwhich have complexity ≤2. (Note that #Ui= (q+ 1)/2 for all i,
hence the error term is On(q−1/2).) 
4. Linearly Disjoint Extensions
The goal of this section is Proposition 13 which asserts that the splitting field of the generic
polynomial is linearly disjoint of any extension that decomposes to extensions of one variable.
This is a technical but crucial point in our proof of Theorem 11.
Definition 12. An extension N/k(t) is called univariate-decomposable or simply decomposable if
there exist finite extensions Ni/k(ti), i= 1,...,n, such that N=N1···Nn.
Proposition 13. Let qbe an odd prime power. Let P(t;x) = xn+t1xn−1+···+tnbe the
generic polynomial of degree nover Fq, let FPbe its splitting field over Fq(t), and let N/Fq(t)be
a decomposable extension. Then FPand Nare linearly disjoint over Fq(t).
We start with some auxiliary lemmas, keeping the notation and assumptions of the proposition.
We also set
tˆ
i= (t1,...,ti−1, ti+1,...,tn)
and we let L=N¯
Fqand Li=Ni¯
Fq(ti), where ¯
Fqis the algebraic closure of Fq. Then, L/¯
Fq(t) is
decomposable, since L=L1···Ln.
Lemma 14. Let Lube a finite extension of ¯
Fq(u). Then there exists α∈¯
Fqsuch that u−αis
not a square in Lu.
Proof. As E=¯
Fq(u)√u−αα∈¯
Fqis an infinite extension of ¯
Fq(u), and Luis a finite exten-
sion, it follows that E6⊆ Lu. In particular, there exists α∈¯
Fq(u) such that √u−α6∈ Lu.
Lemma 15. Assume n≥2. Then DiscxPis not a square in L.

IRREDUCIBILITY OF POLYNOMIALS WITH SQUARE COEFFICIENTS OVER FINITE FIELDS 7
Proof. Recall that L=L1···Ln, where Li/¯
Fq(ti) are finite extensions, i= 1,...,n. For each
i= 1,...,n, let OLibe the integral closure of ¯
Fq[ti] in Liand let OLbe the integral closure of
¯
Fq[t] in L.
We proceed by induction on n≥2. Assume n= 2. Then we have that
DiscxP= Discx(x2+t1x+t2) = t2
1−4t2.
Since L2/¯
Fq(t2) is finite, by Lemma 14 there exists α∈¯
Fqsuch that t2−αis not a sqaure in
L2. Extend the specialization (t1, t2)7→ (2√α, t2) to a homomorphism ϕ:OL→L2. Then,
φ(DiscxP) = −4(t2−α) which is not a square in L2. This implies that DiscxPis not a square in
L, as needed.
Next let n > 2. Consider the polynomials
ˆ
P(tˆn;x) = P(tˆn,0; x) = xn+t1xn−1+···+tn−1x, and
Q(tˆn;x) = ˆ
P(tˆn;x)/x =xn−1+t1xn−2+···+tn−1.
Then, by the definition of discriminants and resultants we have
Discxˆ
P= Discx(xQ) = Res(x, Q)2DiscxQ=t2
n−1DiscxQ.
Set L′=L1···Ln−1. The induction hypothesis for n−1 implies that DiscxQis not a square
in L′, hence Discxˆ
Pis not a square in L′. Since Discxˆ
Pis obtained by substituting tn= 0 in
DiscxP, we deduce that DiscxPis not a square in L.
Lemma 16. The Galois group of Pover Lis Sym(n).
Proof. Let F=¯
Fq(t). Let ˆ
Lbe the normal closure of L/F , then ˆ
Lis also decomposable (since if
L=L1···Ln, then ˆ
L=ˆ
L1···ˆ
Ln), with ˆ
Lithe normal closure of L1/Fq(ti). As the Galois group
ˆ
Gof Pover ˆ
Lis a subgroup of its Galois group over L, it suffices to prove that ˆ
G= Sym(n).
Hence, without loss of generality, we may assume that L/F is normal.
Let FPand LPbe the splitting fields of Pover Fand L, respectively. We set GF= Gal(FP/F )
and GL= Gal(LP/L). Then GF∼
=Sym(n) since Pis the generic polynomial.
Since FP/F and L/F are normal, so is E=FP∩L. Identifying GL= Gal(FP/E) via the
restriction map, gives that GLis a normal subgroup of GF= Sym(n).
LP
FPL
E=FP∩L
F
GL
∼
=GL
GF∼
=Sym(n)
Since all the proper normal subgroups of Sym(n) are contained in Alt(n), it suffices to prove
that GL6≤ Alt(n) to deduce that GL= Sym(n). And indeed, Lemma 15 and Proposition 3 imply
that GL6≤ Alt(n), so GL∼
=Sym(n). 
Proof of Proposition 13.Set K=Fq(t). Let KP,NPand LPbe the splitting fields of Pover
K,N, and L, respectively, and let GK, GN, GLbe the Galois groups of KP/K,NP/N, and
LP/L. We identify these groups with their images under the restriction of automorphism map:

8 LIOR BARY-SOROKER AND ROY SHMUELI
GN∼
=Gal(KP/N ∩KP) and GL∼
=Gal(KP/L ∩KP), so that GL≤GN≤GK∼
=Sym(n). By
Lemma 16,GL= Sym(n), hence GN= Sym(n) = GK. In particular, N∩KP=K, and so N
and KPare linearly disjoint. 
5. Proof of Theorem 11
We first treat a special case, and then deduce the theorem.
Definition 17. Let M > 0. We say that a subset S⊆Fqis of π-complexity ≤Mif it has a
representation as in (4) with m= 1.
There are some obvious connections between the π-complexity and complexity of sets. First, if
the π-complexity is ≤M, then so is the complexity. Second, if Uis of complexity ≤Mthen it is
a union of at most Msets of π-complexity ≤M. Another immediate property that will be used
later is that if Siis of π-complexity ≤Mi,i= 1,...r, then
(11)
r
\
i=1
Siis of π-complexity ≤
r
X
i=1
Mi.
Lemma 18. Let qbe an odd prime power. Let S1,...,Sn⊆Fqbe nonempty and of π-complexity
≤M. Let Pξ∈Fq[x]be a random polynomial as in (1)where ξ1,...,ξnare uniformly distributed
in S1,...,Snrespectively. Then for all s∈ Snwe have that
PhSplitTypePξ=si= n
Y
i=1
isi·si!!−1
+On,M qn−1/2·#S−1
1···#S−1
n,
as q→ ∞.
Proof. Let F=Fq(t) and OF=Fq[t]. Let P(t;x) = xn+t1xn−1+···+tnbe the generic
polynomial, FPits splitting field over F, and OFPthe integral closure of OFin FP. In particlar,
FP/Fqis regular.
By assumption, there exist f11 ,...,f1k1,...,fn1,...,fnkn∈Fq[x] such that ki≤M, 1 ≤
deg fij ≤Mfor all i, j such that
Si=
ki
\
j=1
fij (Fq), i = 1,...,n.
As in Example 10 we may assume that fij(x)−tiis separable in x. We also assume the notation of
Example 10. In particular, as N=N1···Nn, we have that Nis decomposable. By Proposition 13,
Nand FPare linearly disjoint over F. Let Cs⊆Gal(FP/F )∼
=Sym(n) be the conjugacy class of
all permutations with splitting type sand let Ω ⊆Gal(N /F )1be the G0-invariant subset defined
in Example 10. Then, by Corollary 8, (8), and (9), we have
δCheb,1(Cs×Ω) = n
Y
i=1
isi·si!!−1
·#Ω
#GN,1
+On,M q−1/2.
By (10) and since (Qn
i=1 isi·si!)−1≤1, we get that
δCheb,1(Cs×Ω) = n
Y
i=1
isi·si!!−1
·#S1···#Sn
qn+On,M q−1/2.
We identify HomFq(Fq[t],Fq)∼
=Fn
qby the bijection φ7→ α:= φ(t). For φunramified in FPN,
We have OFpN/OF
φ∈Cs×Ω if and only if OFp/OF
φ=Csand ON/OF
φ∈Ω. The former
condition is equivalent to SplitType(P(α;x)) = sby Example 9 and the latter to α∈S1×···×Sn

IRREDUCIBILITY OF POLYNOMIALS WITH SQUARE COEFFICIENTS OVER FINITE FIELDS 9
by Example 10. Since there are at most On,M (qn−1) many ramified φ, we deduce that
n
Y
i=1
isi·si!!−1
=δCheb,1(Cs×Ω)
#S1···#Sn/qn+On,M (qn−1/2#S−1
1···#S−1
n)
=#{α∈Fn
q: SplitType(P(α;x)) = sand α∈S1× · ·· × Sn}
#S1···#Sn
+On,M (qn−1/2#S−1
1···#S−1
n).
This finishes the proof as the coefficients Pξare sampled uniformly from S1× · · · × Sn.
Proof of Theorem 11.For each i= 1,...,n, the set Uiis of complexity ≤M. This means that
there exists mi≤Mand there exist Si1,...,Simi⊆Fqof π-complexity ≤Msuch that Ui=
Smi
j=1 Sij .
If we set
F=nˆ
S=S1j1× · ·· × Snjn1≤ji≤mio,
then ˆ
U:= U1× · ·· × Un=[
ˆ
S∈F
ˆ
S.
Let Ebe the event that SplitType(Pξ) = s. Also, for simplicity we identify subsets ˆ
S⊆Fn
qwith
the event that ξ∈ˆ
S. By the inclusion-exclusion principle we get
P[E] =
#F
X
i=1
(−1)i−1X
ˆ
S1,..., ˆ
Si∈F
distinct
P"E∩
i
\
k=1
ˆ
Sk#
(12)
To ease notation, we fix for a moment one tuple ( ˆ
S1,..., ˆ
Si) of distinct elements of F, and write
ˆ
S=Ti
k=1 ˆ
Sk. Since ˆ
S⊆ˆ
Uand ξis uniform on ˆ
U, we have that
(13) Phˆ
Si=#ˆ
S
#ˆ
U.
Also, if we condition on ξto be in ˆ
S, then it distributes uniformly on ˆ
S. Hence, we may apply
Lemma 18, noting that by (11), the π-complexity of each factor of ˆ
Sis ≤#F · M=OM(1) and
using (13):
(14) PhE∩ˆ
Si=PhEˆ
SiPhˆ
Si=C·Phˆ
Si+On,M (q−1/2#ˆ
U−1)).
where C= (Qn
i=1 isi·si!)−1. Plugging (14) into (12) and noting that the sum in the latter has
On,M (1) terms, we deduce that
P[E] = C
#F
X
i=1
(−1)i−1X
ˆ
S1,..., ˆ
Si∈F
distinct
P"i
\
k=1
ˆ
Sk#+On,M q−1/2#ˆ
U−1=C+On,M q−1/2#ˆ
U−1,
where in the last equation we used the inclusion-exclusion principle for the event ˆ
U.
6. Examples of large size sets of bounded complexity
In Theorem 2 and Theorem 11, the error term may be larger than the main term. This happens
if #Ui≪Mq1/2for at least one i. One the other hand, if #Ui≫Mqfor all i, then the error term
is of size OM(q−1/2) and it tends to zero as q→ ∞.
Let Ube a set of complexity ≤M. The goal of this section is to show that if #U6≫Mq, then
#U≪M1, and to give a sufficient condition for #U≫Mq.
Since Uis a union of sets Sof π-complexity ≤M, we restrict the discussion to the latter sets.
Proposition 19. Let S⊆Fqbe a set of π-complexity ≤M. Then either #S≪M1or #S≫Mq.

10 LIOR BARY-SOROKER AND ROY SHMUELI
Proof. We use Example 10 and its notation, taking n= 1, t=t1=t,πj=πij . Moreover, we
identify φ:Fq[t]→Fqwith φ(t)∈Fq. For an unramified φ:Fq[t]→Fqwe have that φ(t)∈Sif
and only if ON/OF
φ∩π−1
j(Hj)6=∅.
Hence if Ω ∩GN ,16=∅, then the assertion follows from (9). Otherwise, φ(t)6∈ S. In the latter
case, only ramified φmay satisfy φ(t)∈S, and hence #S=OM(1). 
Remark 20.The proof of Proposition 19 gives that for large qwe have #S≫Mqif and only if
Ω∩GN,16=∅.
We give a geometric sufficient condition for #Sto be large.
Proposition 21. Let M > 0, let k≤M, let fj∈Fq[x]be polynomials of positive degree ≤M,
j= 1,...,k. Let Cj={fj(x)−t}and πj:Cj→A1the projection πj(x, t) = t. Assume that the
fiber product D=C1×A1· ·· ×A1Ckis absolutely irreducible. Then,
#
k
\
j=1
fj(Fq)≫Mq.
Proof. Let π:D→A1be the natural pro jection map. Then, deg π=Qdeg πj=Qdeg fj≪M1.
Since πfactors through πj, for each j, we have π(D(Fq)) ⊆Tjπj(Cj(Fq)) = Tjfj(Fq). It
remains to prove that π(#D(Fq)) ≫Mq. And indeed, by the Lang-Weil estimates, #D(Fq) =
q+OM(q−1/2). Hence, #π(D(Fq)) ≥1
deg πq+OM(q−1/2)≫Mq.
7. Comparison with Entin’s theorem
For a set ˆ
S⊆Fn
q, the irregularity of ˆ
Sas defined in [Ent21, eq. (4)] is:
(15) irregˆ
S=qn
#ˆ
SX
β∈Fn
qc
1
ˆ
Sβ.
Let S={α2|α∈Fq}. We compute irreg(Sn) and show that
(16) irreg(Sn)≥(√q−1)n.
This implies that the error term in Entin’s theorem (3) is ≫q(n−1)/2, hence it does not imply
Theorem 1.
Computation of (16).By [Ent21, Lemma 5.1], irreg(Sn) = irreg(S)n. Hence, its suffice to show
that irreg(S)≥√q−1.
Consider the quadratic multiplicative character χ2:Fq→Cdefined by
χ2(α) = 




1 if ∃β∈F×
qsuch that β2=α,
0 if α= 0,
−1 otherwise.
So for α6= 0 we have that
1
S(α) = (1 + χ2(α))/2. Thus, for β∈F×
q,
c
1
S(β) = 1
qX
α∈Fq
1
S(α)eq(αβ) = 1
qX
α∈F×
q
1 + χ2(α)
2·eq(αβ) + 1
q
=1
2qX
α∈F×
q
eq(αβ) + 1
2qX
α∈F×
q
χ2(α)eq(αβ) + 1
q.
By orthogonality of characters, Pα∈F×
qeq(αβ) = −1. By the classical theory of Gauss sums (see
e.g. [IR90, Propostion 8.2.2]), X
α∈F×
q
χ2(α)eq(αβ)=√q.

IRREDUCIBILITY OF POLYNOMIALS WITH SQUARE COEFFICIENTS OVER FINITE FIELDS 11
Thus by the reverse triangle inequality, we have that
(17) c
1
S(β)≥1
2√q−1
q=√q−1
2q.
Finally, since #S= (q+ 1)/2, we substitute (17) in (15) to get that
irreg(S) = 2q
q+ 1 X
β∈Fqc
1
S(β)
=2q
q+ 1 X
β∈F×
qc
1
S(β)+2q
q+ 1 ≥2q
q+ 1
(q−1)(√q−1)
2q+2q
q+ 1
=√q−1 + 2q−2√q+ 2
q+ 1 ≥√q−1.
References
[ABSR15] J. C. Andrade, L. Bary-Soroker, and Z. Rudnick. Shifted convolution and the Titchmarsh divisor prob-
lem over Fq[t]. Philos. Trans. Roy. Soc. A, 373(2040):20140308, 2015.
[AGO+21] T. C. Anderson, A. Gafni, R. J. Lemke Oliver, D. Lowry-Duda, G. Shakan, and R. Zhang. Quantitative
hilbert irreducibility and almost prime values of polynomial discriminants. Int. Math. Res. Not. IMRN,
2023(2021):2188–2214, 2021.
[BBSR15] E. Bank, L. Bary-Soroker, and L. Rosenzweig. Prime polynomials in short intervals and in arithmetic
progressions. Duke Math. J., 164(2):277–295, 2015.
[Bha21] M. Bhargava. Galois groups of random integer polynomials and van der Waerden’s conjecture. arXiv
preprint arXiv:2111.06507, 2021.
[BSG24] L. Bary-Soroker and N. Goldgaber. Full Galois groups of polynomials with slowly growing coefficients.
arXiv preprint arXiv:2404.13559, 2024.
[BSK20] L. Bary-Soroker and G. Kozma. Irreducible polynomials of bounded height. Duke Math. J., 169(4):579–
598, 2020.
[BSKK23] L. Bary-Soroker, D. Koukoulopoulos, and G. Kozma. Irreducibility of random polynomials. Invent.
math., 223:1041–1120, 2023.
[BV19] E. Breuillard and P. P. Varj´u. Irreducibility of random polynomials of large degree. Acta Math.,
223(2):195–249, 2019.
[CD20] S. Chow and R. Dietmann. Enumerative Galois theory for cubics and quartics. Adv. Math., 372:107282,
2020.
[CD23a] S. Chow and R. Dietmann. Enumerative Galois theory for number fields. arXiv preprint
arXiv:2304.11991, 2023.
[CD23b] S. Chow and R. Dietmann. Towards van der Waerden’s conjecture. Trans. Amer. Math. Soc., 376:2739–
2785, 2023.
[Che63] R. Chela. Reducible polynomials. J. Lond. Math. Soc., s1-38:183–188, 1963.
[Coh72] S. D. Cohen. Uniform distribution of polynomials over finite fields. J. Lond. Math. Soc., s2-6:93–102,
1972.
[Die13] R. Dietmann. Probabilistic Galois theory. Bull. Lond. Math. Soc., 45(3):453–462, 2013.
[Ent19] A. Entin. Monodromy of hyperplane sections of curves and decomposition statistics over finite fields.
Int. Math. Res. Not. IMRN, 2021(14):10409–10441, 2019.
[Ent21] A. Entin. Factorization statistics of restricted polynomial specializations over large finite fields. Israel
J. Math., 2021(242):37–53, 2021.
[Gal73] P. X. Gallagher. The large sieve and probabilistic Galois theory. In Analytic number theory, volume Vol.
XXIV of Proc. Sympos. Pure Math., pages 91–101. Amer. Math. Soc., Providence, RI, 1973.
[Ha16] J. Ha. Irreducible polynomials with several prescribed coefficients. Finite Fields Appl., 40:10–25, 2016.
[IR90] K. Ireland and M. Rosen. A Classical Introduction to Modern Number Theory, volume 84 of Graduate
Texts in Mathematics. Springer New York, NY, second edition, 1990.
[Kno57] HW. Knobloch. Die seltenheit der reduziblen polynome. Jahresber. Dtsch. Math.-Ver., 56(1):12–19,
1957.
[Kub09] G. Kuba. On the distribution of reducible polynomials. Math. Slovaca, 59(3):349–356, 2009.
[LW54] S. Lang and A. Weil. Number of points of varieties in finite fields. Amer. J. Math, 76(4):819–827, 1954.
[May19] J. Maynard. Primes with restricted digits. Invent. math., 217:127–218, 2019.
[Mos17] E. Moses. Irreducible polynomials with varying constraints on coefficients. arXiv preprint
arXiv:1712.04051, 2017.
[Pol13] P. Pollack. Irreducible polynomials with several prescribed coefficients. Finite Fields Appl., 22:70–80,
2013.

12 LIOR BARY-SOROKER AND ROY SHMUELI
[Por19] S. Porritt. Irreducible polynomials over a finite field with restricted coefficients. Canad. Math. Bull.,
62(2):429–439, 2019.
[Ros02] M. Rosen. Number Theory in Function Fields, volume 210 of Graduate Texts in Mathematics. Springer
New York, NY, first edition, 2002.
[vdW36] B. L. van der Waerden. Die seltenheit der reduziblen gleichungen und der gleichungen mit affekt.
Monatsh. Math. Phys., 43(1):133–147, 1936.
[Zyw10] D. Zywina. Hilbert’s irreducibility theorem and the larger sieve. arXiv preprint arXiv:1011.6465, 2010.
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv
Email address:barylior@tauex.tau.ac.il
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv
Email address:royshmueli@mail.tau.ac.il