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Primitive elements and polynomials with arbitrary trace

July 1990
Discrete Mathematics 83(1):1-7

DOI:10.1016/0012-365X(90)90215-4

Source
DBLP

Authors:

Stephen D Cohen

University of Glasgow

With one non-trivial exception, GF(qn) contains a primitive element of arbitrary trace over GF(q).

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Discrete Mathematics 83 (1990) l-7

North-Holland

PRIMITIVE ELEMENTS AND POLYNOMIALS WITH

ARBITRARY TRACE

Stephen D. COHEN*

Department of Mathematics, Uniuersity of Glasgow, Glasgow Cl2 SQW, Scotland, UK

Received 29 May 1987

Revised 10 February 1988

With one non-trivial exception, GF(q”) contains a primitive element of arbitrary trace over

GF(q).

1. Introduction

Let F, = GF(q) denote the finite field of prime power order q. The presence in

F2” of primitive elements (or roots) having trace 1 over F2 has been useful in

coding theory, see [2] and [8, Ch. 41. Although, in this instance, existence is

guaranteed by Davenport’s general construction of a primitive normal basis [5], in

resolving a problem raised in [8], Moreno [7] provided a simple direct proof.

In this paper it will be shown that the above result is a special case of a much

more general one (not derivable from Davenport’s theorem) which states,

essentially, that in any proper extension F,” of F, there exists a primitive element

with arbitrary trace t in F4. (Recall that T(g), the trace of 5 in F4” over F4, is

given by

T(E) = ij + i$” + 5”’ + . . * + pm’

and is automatically a member of F4.) Equivalently, there exists a primitive

polynomial f(x) of degree it over Fg with trace t (so that f(x) =x” - tx”-’ + - . -).

In this sense, it is clear that, if it = 2, no binomial x2 - a (where a E F4) can be

primitive (except when q = 3); thus we must disallow t = 0 in this case. Less

evidently, it can be seen by means of tables that none of the twelve primitive

polynomials of degree 3 over F4 (see [6, Ch. lo] have zero trace either. That these

are the sole exceptions is established in the following theorem.

Theorem 1. Let n 2 2 and t be an arbitrary member of F4 with t # 0 if n = 2 or if

n = 3 and q = 4. Then there exists a primitive element in F4” with trace t.

Equivalently, there exists a primitive polynomial of degree n over F4 with trace t.

*This paper was written during a visit by the author to the University of the Witwatersrand,

Johannesburg.

0012-365X/90/$03.50 0 1990- Elsevier Science Publishers B.V. (North-Holland)

2 S.D. Cohen

Theorem 1 derived from following result; extends the theorem of

where we n =

Theorem 2. Let n and t be as in Theorem 1 and let {wl, . . . , o,} be any basis of

F$ over F,. Then there exists elements al, . . . , a,_, in Fg such that alwl + . . - +

a,_,~,_, + tw, is a primitive element of F4”.

Theorem 2 can be cast more geometrically as follows.

Theorem 3. Suppose n 3 2 and F4” is regarded as an n-dimensional afine space

over F4. With the exception of hyperplanes through the origin when n = 2 or when

n = 3 and q = 4, every hyperplane contains a point corresponding to a primitive

element of Fg”.

2. Estimate for a character sum

In this section t is a non-zero element of F4 while throughout { wi, . . . , w,} is a

basis of F4” over F,. As shorthand we write a for (a,, . . . , a,_,) E Fi-’ etc., and

put 0 for (wi, . . . , w,-J and (I - o for the “inner product” alwl + . . . +

an-lwn-l.

Lemma 1. For all E in F4” but not in F4 there are q”-’ solutions (x, y) E Fr-’ of

the equation

(x * 0 + tw,)l(y - c-0 + twn) = 5. (2.1)

Proof. Suppose Ewi = Cyzl a,wt, i = 1, . . . , n (aij E F4). Then (2.1) holds if and

only if

E( i: Yiwi) = 2 (2 Yiaij) Wj = i: xjwj, %I = Yn = t7

i=l j=l

=x1 -a,,y, -a - . - 4-l lh-I = a, 1

* * X,-l -aln-lyl -. . . - an--ln--lhl = ann-lt

-aInyl - . . . - an-lnx-l = (arm - l>t.

Obviously these n equations in 2n - 2 unknowns have q”-’ solutions unless

ajn=O, j=l,..., n - 1 which, however, implies that &J = arm, a contradiction,

because 5 $ F4. 0

Primitive elements and polynomials with arbitrary trace 3

Now let x be a multiplicative character of F,. and define

S(x) = c x(a * w + mz),

where the sum is over all a E FG-‘. Also let Q = (q” - l)/(q - 1). We obtain an

estimate for IS(x)1 whose value depends on whether or not the order of x divides

Lemma 2. Suppose x is a non-principal character of order d(>l), where

d ) q” - 1. Then

Proof. Let 2 be the conjugate of x. Then

lW12 = ww

= B z& x((a . CfJ + WJlvJ -0 + fwt))

, 4

= 4 n-2 ,& x(E) + 4”-Ml),

” 4

by Lemma 1 and the fact that (a - o + tw,)/(b - OJ + &on) lies in F4 only if u = b

whereupon its value is 1. Hence

lw12 = -F & x(E) + e-l.

Eq (2.2)

Now the sum in (2.2) is zero if the restriction of x to Fg is non-principal, i.e. if

d # Q and otherwise (when d 1 Q), the right side of (2.2) has the value

-q”-*(q - 1) + q”-’ = q”-*. The result follows. Cl

Now let e be a divisor of q” - 1 and define N(e) to be the number of elements

of the form E = a - w + to, (i.e. on a given hyperplane) for which E # 0 and the

integer defined by (q” - l)/(order of 5) and e are co-prime. In fact it suffices,

except in one instance, to examine the particular case in which e = q” - 1, when

N(q” - 1) is simply the number of primitive elements on the hyperplane. In terms

of character sums (see [3]), we have

(2.3)

where @ and p are the functions of Euler and Mobius and CXCmodd) denotes a sum

over all characters of order d. Focusing on the case e = q” - 1, write N(q, n) for

((4” - l)/#(q” - l))N(q” - 1). In similar vein to [7] it is enough to prove that

N(q, n) is positive. To this end we estimate iV(q, n): evidently, an analogous

estimate holds for any N(e). The result follows immediately from (2.3) and

Lemma 2.

~z~a~!p 6q ~13 b 11” 103 paqsqqwa aq um (z.E) ‘~aaoa.~on .1ClaAyDadsar

‘cl< b 10 s < b uaqM paysyes 6Iawpatuuq s! (z’E) uaql ‘p JO E = A 3’ ‘laqwd

‘z IO T = A uaqM sploq (z-s) ~eq~ aas liI!sea um aM ‘z = b uaqm s = 1 ~eq~ In3pu!m

(ZX) j_b(r - n) + ,z - ,z < b

31 aprs pueq-)q2@ ag!sod seq (py,) ‘uoye~ou s!q~ pavmg

(IX) .(Epow)T=b3! ‘T+s---l =

‘(E Pow) 1 f b 3! ‘s- 1

I I

‘dlJeal3 ‘uaqL

'(1 - b)M = 1 pue (r<)(a)M = S ‘(I~)(I - Eb)M = 1 al!sIM ‘dJ!Aalq

_IO~ *(( 1 - b) + d pm) (9 pour) 1 e d .IO ((c pow) 1 G b asw qxqm u!) E = d Jaqlfa

‘uaql I+ b + ,b = 0 sapyp d at.u!ld c 3! wq~ ysq aloN *c = u ~eq~ asoddns

aM .Asea &?u~sea.mu! sauroDaq uogmy~~aa aq$ q3IqM .1a~3c ‘c = 24 uaqM aAgDa33a

6llmauaZ s! c euma~ leq$ yDaq:, 01 paqnbal s! llo33a aqt 30 ylnq aql ‘d@uydms

JON ‘0 f 3 pm E =z u wql aumsse um ahi ql.Io3amaq ‘aAoqc aq1 ur0.g

*c t.ualoaqL uoys;raA aqi u! uoyqmo:,

pa.zrnbal aql saqsqqe$sa qmqM tuatuala ah!)!ur!Id I? s! 2 D ‘“J 3 0 luau.IaIa

alqe$!ns e “03 1Cl.reaI3 .a 01 awild s! (2 30 Japlo)/(T - ,b) leql OS ‘0 01 au.yld

s! (waura~a aAyw!Jd due 01 IDadsal q$IM) xapu! asoqM 2 waurala ue sup?v~o~

aDeds-u augc u! Z&JO azg z&o+ auelhaddq haha Jeql ‘IxawoD mo u! ‘sagdur!

q3!qM 0 lap10 30 luaurala ue suyuo~ (E c u alaqM) “d .raAo azezds-u anyafold

u! aueldladliq haaa ‘p = b uaqM IdaDxa ([ 11 ‘aldwexa 103 ‘aas) Klwanbasuo3

‘A olnpour sla8auq 30 dnol% aAg!ppe aql 30 lomauak? tz OS pm A 01 awgd

anpjsal e suguo:, las azmaIa33!p-(-f ‘y ‘A) mph halla ‘slas amaJaJJ!p-(1 ‘s ‘1~)

OMI u10.13 mde ‘]??q) syasse s!qA .(suo!~cln3le3 pue suo!)elapIsuoD a]eyap atuos

su!eu~o:, u~e8e 3oold asoqM) [p] 30 T. uraloaqL 111013 &sea SMOIIOJ $Insal aq$ $eq$

MOqS MOU aM ‘0 = 1 ‘&iXIoyppF? ‘31 ‘C * U )eql asoddns alo3araql km aM ‘z = u

uaqM sploq z uraloaqL (s!sdpme In3a.m 30 aurowno aq,) [c] 30 ~‘1 uraIoaqL 68

wanzdde ssal ‘IaAaMoq ‘s! (p-z)

30 6xcga leraua% aqJ ‘asm leql u! 1 uraloaqJ spIa!A lilycynb haA pue [L] 30 pua

aqy vi? oualoH hq paqsqqelsa uo!l!puo:, aql LlaspaJd s! qXqM I - (1 - ,Z)M < &

pap!Aold aAg!sod s! (U ‘z)~ ieql saqduq (p-z) uaql z = b uaqM Jeql ymual aM

(P.Z> .z,cz-ujb(I - (a)M) - ,,+u,b((~)M - (I - ub)M) - I-ub * t” ‘b)N

sau+d 13uys~p Jo laqtunu at# salouap M a.iayM (,,,z = (a)M lay ‘E: sumaq

Primitive elemenis and polynomials with arbitrary trace 5

verification except when q = 11 in which case q3 - 1 = 1330 = 2.5.7.19, r = 4 and

s = 2. This value is investigated separately at the end.

To assist the discussion of the values r 2 5, introduce, for each positive integer

m, integers A(m) and B(m) defined, respectively, as the product of the first m

primes and the first m primes which are congruent to 1 modulo 6. We have

q’A(t)’ B(s),

’ 5 (3B(s - l), if q + 1 (mod 3),

if q = 1 (mod 3). (3.3)

Hence if m satisfies

A(m) (3.4)

we can conclude that (3.2) is valid for t 3 m (even when s = 1 because q is an integer).

If r = 5, then (3.4) holds with m = 3 and so we can assume t c 2. In turn, this

implies s 2 3 (by (3.1)), Q 2 B(3) = 1729 (by (3.3)) and q > 41 which is stronger

than (3.2).

If r = 6, then (3.4) holds with m = 4 and we can take t c 3. Indeed if t = 3 and

q $1 (mod 3) then q 2 71 which implies (3.2). Otherwise s 2 4 (by (3.1)) and

hence Q 3 5187 (by (3.3)) g

a ain, coincidentally, yielding q > 71.

The treatment of the cases r = 7 and 8 is similar to that of r = 5 and 6,

respectively. Noting that B(4) = 7.13.19.31= 53599, we obtain q > 230 (when

r=7)andqa771andq>4OO(whenr=8).

For r 2 9 we use the following facts which can be established by induction for

m 3 5, namely,

A(m) 3 2”“, B(m) > 6”(m + 1)!/3 > 24m (m 2 5). (3.5)

Selecting m = [b(r + l)] (employing integral part notation), we deduce from (3.5)

in the first place that t < m and so s sm. Again by (3.5), were (3.2) to be false,

we would have

4m 2 22’ 2 (q + 1)’ > Q > B(m) > 24m,

a contradiction. With the exception q = 11 noted above, this completes the proof

for n = 3.

As IZ increases the working becomes significantly easier. Suppose for instance

n = 4. If q is odd, then 16 divides q4 - 1 and no other special considerations are

necessary. For example, if w(q4 - 1) = 4 then q4 - 12 16.3.5.7 = 1680 which

already implies qs > 16; consequently N(q, 4) is positive by Lemma 3. Similarly,

if q is even, then q4 - 12 3.5.7.11= 1155 and qs > 14 and this is enough.

When n = 5 it suffices to prove that q2 > W(q5 - l), a task which is facilitated by

the fact that necessarily p = 1 (mod 10) for all p(>5) dividing Q. Larger values of

IZ are clearly a formality and we omit further details.

From the above for the completion of the proof of Theorem 2 the only lack is a

demonstration that N(11, 3) is positive. In this case q - 1 = 10 and Q = 133. For

any e let M(e) denote the set of points on the relevant hyperplane (here a line)

6 SD. Cohen

which are dth powers for no divisor d(>l) of e. Then, of course, ]M(e)l = N(e)

always, while, in particular

M(lO) r-l M(133) = M(1330).

Hence

N(lO) + N(133) - N(1330) 6 N(1) = 121

and so

N(1330) s N(lO) + N(133) - 121.

On the other hand by (2.3) and Lemma 2 we have

N(lO) 2 $(121- 33) = 35.2,

N(133) 2 $$(121- 3Vll) > 90.1.

Thus N(lO) 2 36, N(133) 3 91 and N(1330) 2 6 by (3.6). 0

(3.6)

4. Proof of Theorem 1

We require a simple lemma.

Lemma 4. There exists a basis { ol, . . . , w,} of Fqn over Fq with

T(q) = 0, i = 1, . . . , n-l and T(w,)=l.

Proof. Since T(F,.) # 0 we can pick E E Fq” with T(E) #O. Put w, = T(E)-‘&

Then let {o;, . . . , WA-~, co,} be any basis extending o, and define wi = ol -

T(of)o,. 0

The proof of Theorem 1 is now readily completed as follows. Let

(01,. . * > o,} be the basis constructed in Lemma 4. From Theorem 2 let y be a

primitive element of the form alwl +. * * + an-lm,_l + tw,. Then T(y) = t as

desired. Cl

Note added in proof. Independent partial treatments of Theorem 1 (but not

Theorems 2 and 3) have recently appeared. 0. Moreno has dealt with the case

n = 2, t = 1 in his paper “On the existence of a primitive quadratic of trace 1 over

GF(p”)” (J. Combin. Theory Ser. A 51 (1989) 13-21). In their paper “On

primitive polynomials over finite fields” (J. Algebra 124 (1989) 337-353), D.

Jungnickel and S.A. Vanstone have shown that Theorem 1 holds for all but a

finite number of values of 9 and n. In particular, for t # 0, they identify an

explicit list of possible exceptions (all with n = 2). These papers contain working

akin to the present article but it seems that ideas on a level with those of [3] and

[4] would be needed for completion.

Primitive elements and polynomials with arbitrary trace 7

References

[l] L.D. Baumert, Cyclic difference sets, Lecture Notes in Mathematics 182 (Springer, Berlin,

Heidelberg, New York, 1971).

[2] E.R. Berlekamp, Algebraic Coding Theory (McGraw-Hill, New York, 1968).

[3] S.D. Cohen, Primitive roots in the quadratic extension of a finite field, J. London Math. Sot. 27

(2) (1983) 221-228.

[4] S.D. Cohen, Generators in cyclic difference sets, J. Combin. Th, Ser A 51 (1989) 227-236.

[5] H. Davenport, Bases for finite fields, J. London Math. Sot. 43 (1968) 21-39.

[6] R. Lid1 and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications 20

(Cambridge, 1983).

[7] 0. Moreno, On primitive elements of trace equal to 1 in GF(2m), Discrete Math. 41 (1982) 53-56.

[8] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error Correcting Codes (North-Holland,

Amsterdam, New York, Oxford, 1977).

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