Matrix powers over finite fields

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Internat. J. Math. & Math. Sci. VOL. 15 NO. 4 (1992) 767-772

767

MATRIX

POWERS OVER FINITE FIELDS

MARIAT.ACOSTA-DE-OROZCO

Department

ofMathematics

Penn StateUniversity

Beaver

Campus

Monaca,

Pennsylvania15061

and

JAVIERGOMEZ4.’ALDERON

Department

ofMathematics

Penn State

University

New

Kensington

Campus

New Kensington, Pennsylvania15068

(Received May

23,1991andinrevisedform

May

18,

1992)

ABSTRACT. LetGF(q)denote thefinitefield of orderq

pe

withpodd. LetM denote the ring of 2x2matrices with entries in GF(q). Let,denoteadivisorofq-1 andassume2<,and4does not divide

,.

In

this paper,weconsider theproblemof determining the number of,- throots in Mofa matrix BEM. Also, asarelatedproblem,weconsidertheproblemofliftingthe solutions ofX2 B overGaloisrings.

KEY

WORDS AND PHRASES.

Finitefieldsandmatrixpowers. 1991

AMS SUBJECT

CLASSIFICATION CODES.

Primary 15A33.

1.

INTRODUCTION.

Let

GF(q) denote the finitefield of order q

p"

with p odd.

Let

M denote the ring of2x2 matrices withentries inGF(q)..Let ndenoteapositive divisor ofq-1:

In

this paper, weconsider theproblem of determiningthe number N N(n,B) ofn-throots inM ofamatrix BEM; i.e., the number of solutionsin Mof the equation

Xn=B

(1.1)

Our present work generalizes a recent paper ofDonovan

[I]

in which thequadratic equation

x

2 _{B is}_solved_over_{the ring}_M.

As a related problem, wealso considerthe problem oflifting solutions of equation

(1.1),

for

n 2, over Galois rings. The Galois ringoforder

prm,

denotedby GR(pr,rn), can beobtained asa

Galois extensionof

Zpr

of

degree

m. The reader canfindfurtherdetails about Galois rings in the

reference

[4].

IfB denotes a scalar matrix, a multiple ofthe identity matrix, then equation

(1.1)

is called

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768 M.T. ACOSTA-DE-OROZCO AND J. GOMEZ-CALDERON

n 2and Bdenotes theidentity matrix, thenthe solutions of

(1.1)

arecalled "involutorymatrices".

Involutory matrices over either a finite fieldor a quotient ring of the rational integers have been extensivelyresearched,withadetailedextension toall finitecommutativerings given by McDonald

in

[51.

2. OVERFINITE FIELDS.

Let GF(q) denote the finite field of order q

pe

with p odd. Let M denote the ring of 2x2 matrices with entries in GF(q)and let GL denoteitsgroup ofunits.

For

each Bin Mlet S(B) and

[B] denote,respectively,thestabilizerandthe conjugateclassofBdefinedby

and

S(B) {AEGL:AB BA}

(2.1)

Thus

[B] {ABA-

I:A

eGL}.

(2.2)

I[B]I [GL:S(B)].

Nowforthe purpose of the present workwewill needthefollowingstabilizers:

(2.3)

(i)

S((: Oa))--

GL(q

(ii)

Wenowgiveaseriesof lemmas fromwhichourmainresult, Theorem6,will follow.

LEMMA

1.

Assume

Tn BforsomeTandsomenon-scalarBinM. Then$(T)

PROOF.

Since B is non-scalar, the minimal polynomial of T is a quadratic polynomial

IT(Z)

x2

+

az

+

b. Therefore, B Tn dT

+

eI for some constants e

ad

0#d in GF(q). Thus,

$(T) S(B).

LEMMA

2. Ifn_>2then the number ofmatrices T in MsothatTn 0is

q2.

PROOF. Tn 0 if andonlyiftheminimal polynomialof T is eitherxorz

2.

Hence,

Tn_--0_if

=.

i.. to

(?

T,e

I{T M:Tn 0}] I[A]I

+

I[B]I [GL:S(A)I

+

[GL:S(B)I

/q(q

1)(q2 1)/(q2

q)

q2.

LEMMA

3.

Let

2<ndenote adivisor ofq-1 and assumethat 4 does notdivide n.

For

each rin

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IqATRIX POWERS OVER FINITE FIELDS 769

a)

n

+

(q2

q)(n-1)n/2 if rEGF(q)n

b)

(q2_q)n/2

if r GF(q)n butr2$GF(q)n

c)

0 if r2 GF(q)n

PROOF.

Let wdenoteaprimitive element ofGF(q) andwriter wmfor someinteger _<m<q- 1.

Then Tn=diag(r,r) if and only if the minimal polynomial of T divides

I(z)=zn-w

m.

Now.

if

D (n,m)denotes thegreatestcommondivisorofnandm,thenweobtain

D-1

+m/D)

H

(

zn/D-

w(q

1)i/D i=0

D-1

i=0

We

alsosee that w(q-1)i/D+rn/D does not

belong

to GF(q)s for every odd prime factorsof n/D. Therefore, by

[3,

ch.

VIII,

Th.

16],

hi(z

is irreducibleover GF(q) for all i. Thus, n/D 1,

n/D 2 orthereare nomatricesTsothatTn diag(r,r).

CASE

1: n/D 1. Then ndivides mand

T"=

diag(r,r)ifand only ifthe minimalpolynomial

of Tis either z-aor (z-a)(z-b)where aand bdenotetwo distinct roots inGF(q)ofthe equation

zn=r. Hence, Tn=diag(r,r) if and only if T is similar to either A=diag(a,a) or B =diag(a,b).

Therefore,

I{T M: Tn diag(r,r)} =hi[all

+f

q(q-l(q

2-1)

(q-1)2

n

+

(q2

+

q)(n 1)n/2

CASE

2: n/D 2. Then n/2dividesmandTn diag(r,r)ifandonlyif the minimalpolynomial

ofT is aquadratic irreduciblepolynomialoftheformz2- wherecdenotesa rootof the equation

(

c)Therefore,

zn/2 r. Thus,Tn diag(r,r)ifandonlyif Tissimilar to A 0

I{T M:Tn_--diag(r,r)} q(q-

1)(q2-

1)n

(q-

,)(2)

ifr

f

GF(q)nbutr2 GF(q)

n.

LEMMA

4. IfTn diag(h,k) with h#k, thenT diag(r,s)forsomerandsin GF(q).

PROOF.

Let

f(z)=z2+az+b denote the minimal polynomial of T.

So,

T2=-aT-hi

and cT

+

el diag(h,k) forsomecand in GF(q). Therefore,T diag(r,s)forsome andsinGF(q).

LEMMA

5.

A

non-scalar 22 diagonalizablematrix over GF(q)is a n-th power in M ifand onlyifitseigenvalues, necessarily distinct,axen-thpowersin GF(q).

PROOF.

Assume T to be non-scalar and diagonalizable so that for some matrix P in GL,

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770 M.T. ACOSTA-DE-OROZCO AND J. GOMEZ-CALDERON T P-

ldiag(h,k)P

P-

l{diag(r,s))nP

(P-

ldiag(r,

slP)

n.

Conversely, suppose T Nn and T is diagonalizable. Say

P-ITp

=diag(h,k) where h k are the eigenvaluesof T. Hence

diag(h,k) P-1Tp P-1Nnp (P-

1Np)n.

Therefore, byLemma4, P-1Np diag(r,s)with rn hand/:n s.

THEOREM 6. Let B denotean element ofM. Let ndenote a divisorofq-1.

Assume

2<n

and4does not dividen. ThenBhas

(a)

morethan n2n-throots in Mifandonlyif B rlforsomerin GF(q)sothat r2EGF(q)

n.

(b)

exactlyn2distinct n-throots in M ifandonlyifBhas unequalnonzeroeigenvalueswhichare n-thpowersinGL(q).

(c)

at mostndistinctrootsin M,otherwise.

PROOF. IfB rlforsomerin GF(q),then, by

Lemma

3,Thas

(i)

morethan

n2n

throots if andonlyifr2EGF(q)nand

(ii)

zeron-throotsif andonlyif

,.2

q GF(q)n. WenowassumethatT isnon-scalar.

CASE 1: B diagonalizable. Then by Lemma 5, B is a n-th power in M if and only if its

eige,nvalues,necessarilydistinct,aren-thpowersin GF(q). Therefore,Bhas exactly

(iii)

n2 distinct n-th rootsin M if and onlyifB hasunequal nonzeroeigenvalues which are n-th powers in GF(q) and

(iv)

zeron-throots otherwise.

CASE 2: B non-diagonalizable. Then the minimal polynomials of both B and T are either:

quadraticirreducibleor quadraticperfect square polynomials. We alsoseethat if

Tn=

Bthen the

minimal polynomial of Tis a factor of

IB(z

n) where

IB(Z)

denotes theminimal polynomial ofB.

Therefore, thereare at most npossible minimal polynomial

IT(Z).

Further,

(P-1Tp)n=

Bifand

onlyif P ,q(B). Therefore,since [S(B):S(T)] by Lemma 1, Bhas at mostn distinctn-throots

in M.

3.

LIFTING SOLUTIONS.

Let GR(pr,m) denote the Galois ring of order

prrn

witht’odd.

For

purposesofconstructionand easeof implementation of Galois rings,one canconstructGR(pr, rn) by considering

(zpr)[zl/(I)

where

I

is a monic irreducible polynomial of degree rn_> over the finite field GF(prn) GF(q) with p

prime.

Furtherdetailsconcerningpropertiesof Galois ringscanbe found

ia

the reference

[4].

In

this section, we will consider a special case, n 2, of hfting solutions over Galois rings.

Morespecifically,wewillprovethefollowing

THEOREM 7. Let

M(pr+l,rn)

denote the ring of all 22 matrices with entries in GR(pr+

1,rn).

Let A denote anelement ofM.

Assume

that

,

the reduction ofAmodulo p, is a

(a

bd)

M(pr,m) denoteasolution of

X2=

A (rnod non-scalarinvertible matrix in M(p,m). Let X

c

lr).

Then Xo canbe hfted from

M(lr,m)

to M(Ir+

1,m)

in

(a)

auniqueway if #0.

(b)

q

prn

different ways if either 0 or #0and 0.

(c)

q2

p2m

different waysif #0and 0.

PROOF.

Let

x=(

Y)w

where z,y,z and w areelements of the field GR(p,m)to be specified

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MATRIX POWERS OVER FINITE FIELDS 771

(Xo+

Xpr)2

=_

X2o

_+(Xo

x

₊_XXo)prmod

pr

+

Now,

since

x.

2 Aover GR(vr,m), wecan write

xo

2 A-Cprfor some 22matrixCover the ring

GR(p,m). Hence,

(Xo+Xpr)2 A+(XoX

+

XXo-C)prmod

pr

+

Therefore,(Xo

+

xpr) Aoverthe

tins

GR(pr

+

1,m),

ifand onlyif

overthe fieldGR(p,m);i.e., if andonlyif

XoX

+

XX C

Cl

c2)

whereC= c3 c4

Hence,

wehave to count the number of solutions,in GR(p,m),of thelinearsystem

/ w x

c b 0 2a c

a+d 0 b b c

2

0 a+d c c c

3

c b 2d 0 c

4

(roodp)

or

t z w x

ClC

c(a

+

d) 0 bc be

0 a+d c c c3

0 0 2bcd 2abc

--,

c4 c

1)b

0 0 0 E

E₂

(rood p)

where E =2(a+d)(ad-bc) and

E2=clad+cld2-c2cd-bc3d+c4bC-clbC.

So, since 3 is non-scalar and invertible, E #0. Therefore, a straightforwardinspection of the above last

augmented

matrix

willcompletetheproof ofthe theorem.

REFERENCES

1.

DONOVAN,

T.P.,

Matrix squaresoverGF(q), qodd,

Congressus

Numerantium66

(1988),

113-122.

2.

HODGES,

3.H., Scalar polynomial equations for matricesover afinitefield,

Duke

Math. 3.25

(1958),

291-296.

3.

LANG, S., "Algebra",

Addison-Wesley, Reading, Mass., 1971.

4.

MCDONALD,

B.R.,

"Finite

Rins

withIdentity

",

MarcelBekker, NewYork, 1974.

5.

MCDONALD,

B.R.,

Involutorymatrices overfinitelocal rings, Canadian Journal ofMath. 24