Ie J Mh & Mh Sci Vol 8 No 2 (1985) 373-388 373 ON FUNDAMENTAL SETS OVER A FINITE FIELD YOUSEF ABBAS d JOSEH J LIANG Dee of Mheic Uiveiy of Souh Floid T, Floid 33620 USA (Received Mch 3, 1983) ABSTRACT A iio ove fiie field i defied d ech equivlece cl i couced d eeeed by e clled he fudel e If iiive elee i ued o couc he ddiio ble ove hee fudel e he ll ddiio ove he field c be coued The ube of iio i give fo oe fiie field KEY WORDS AND HRASES Fiie field, iiive olyoil 19B0 MATHETICS SUBJECT CLASSIFICATION CODE 12C99 i INTRODUCTION Thoughou,q will be fixed bu biy ie Le E GF() > 1 d defie A A {, +l, +2, =+ (-l)} If B E A, he A A Defie 2A, 3A (-l)a uch h A {x/x A} 1,2,,- I, hu A A DEFINITION * A D 1 =i,,, Noe h if 8 6 Ae, he A AS" A= {x/x DEFINITION A fo 0,i 2 o, A, +I, A ( C ( + (-l)} A Sice fo evey GF() we hve A A Theefoe -U I= he vlue of i he defiiio c be liied o 0,1,2,,-i DEFINITION DEFINITION A, * A (A -l A,A Sice (Z,+,) hu X { 3 +bl Z* b Z d j E Z } LEMMA ii If 8 EX, he X A--B ROOF If 8 6 A he hee exi 6 Z b 6 Z d J Z uch h 8 j + b Thi ilie 8C -1 o A A8" i field d 0 boh
374 Y ABBAS AND J J LIANG DEFINITION A will be clled Fudel Se Sice A A fo evey E GF(), hee exi le oiive iege <_ A * uch h A fo oe 6 Z I follow h A *A DEFINITION Le u,8 E GF() We defie he elio i GF() 8 iff A AB THEOREM ii Fo Le ii he elio i equivlece elio Thi equivlece elio will iio he field GF() io equivlece cle A d ech cl i eeeed by fudel e DEFINITION A will be clled Fudel Cl _ i Le 8 be iiive elee i GF() Sice, 8 -I i iiive i Z, he fo evey Z hee exi k whee i < k < - i So, k c be deeied eily If he elee of he fudel e A e exeed owe of 8, he c be exeed fo A owe of 8 So o clcule he ddiio ble of A i i ufficie o hve he ddiio ble of A Theefoe, if AI,A2,,A=, e ll he fudel cle i GF(), i will be eough o bule oly he ddiio ble ove A,,A wih eec o 8 o do ll he clculio ove i GF(), If fo oe GF(), i he le oiive iege uch h + b fo oe Z d b Z he i i ue h will be he lle oiive iege fo evey B uch h B B+ b whee b E Z Thi will be how below DEFINITION Le GF(), if i he le oiive iege uch h + b fo oe Z * b Z he i clled he idex of d i he coefficie of d we y h idex wih coefficie If = Z we y h idex 0 wih coefficie i LEMMA 12 If h idex wih coefficie, he evey 8 A h he e idex d he e coefficie ROOF Le B wih idex d B Z Thee exi E Z 6 Z d j Z uch h 8 J + 6 Thi ilie 8 ( j + 6) ( + b) j + 6 + c whee c Z Theefoe _< Bu fo Le ii we hve A A 8 Hece A Theefoe <, which ilie 8 h idex wih coefficie DEFINITION The fudel e A h idex wih coefficie if d oly if i he le oiive iege uch h A A THEOREM 12 If h idex wih coefficie he A h idex wih coefficie d ech fudel e A C h he e idex wih he e 8 A coefficie ROOF Follow fo Le 12 Fo he bove heoe we c defie he idex A o be he idex of y elee o y fudel e icluded i A Now, we w o dicu oe oeie of he idex d he coefficie of he fudel e -i
FUNDAMENTAL SETS OVER A FINITE FILED 375 THEOREM 13 I GF(), if A h idex d coefficie he dfvlde d / I ROOF Le +, 0 < < Sice + b, fo oe be Z we hve -2 -i -2 () + ( -I + ++ l)b d ( ) - + ( + + + l)b So A A which ilie 0 d / i THEOREM 14 Le A be fudel e wih idex d coefficie If * / A ba, b E Z he divide d b ROOF Sice i he idex of A, we hve < Le k +, 0 _< < If A, he hee exi 6, 6 Z uch h + 6 d b+ 6 Bu k, b+ 6 ( ) (ko + 6 ) + 6 k + 6" Thi ilie > Theefoe 0 Hece l d b Noe: Sice ll fiie field of he e ode e ioohic, if i E GFI() hvig idex wih coefficie, i he ige of h idex wih coefficie Thee- 2 GF2() ude ioohi o, he 2 foe; GFl() d GF2() elio hve he e iio wih eec o he equivlece EXAMLE: Le F GF(52) d 8 be iiive elee i F uch h 8 ifie he iiive olyoil idexig olyoil], (x) x 2 + 4x + 2 The field F h wo equivlece cle, Z d A 5 8 {8, 8 + i 822 8 + 2 815 8 + 3 82 817 Sice 86 2, he 4 812 d - 8 + 4 817 So, we hve 8 8228/ 815 82 3 818 Theefoe, 2A 8 {8 7 87 + 2 8 4 87 + 4 =821 87 + 1 88} 3A 0 {019 019 + 3 016 019 + i 0 I0 019 + 4 821 019 + 2 06 4A 8 {813,813 + 4 810,813 + 3 83, 813 + 2 814, 813 + I 85 5 d A 4A 8 8 2 SOLUTIONS OF EQUAl IONS OF THE FORM A A To udy he fudel e i GF() wih idex <, we hve o udy he, x * x oluio of: () x + 6; Z # i, 6 E Z d (b) x + 6; 6 Z i GF() Noe: If 6 0 i (b), he ll he elee of he ubfield GF() will ify (b) We will ow coide he oluio of x x + 6 (21), whee 6 Z, # 1 d 6 Z LEMMA 2 i Equio (2 i) h oluio e Z wih idex oly if divide d i The oof i diec licio of Theoe 13 d Theoe 14 LEMMA 22 I equio (21), fo y x + -1 we hve y y ROOF )=x y (x + + x + 6 + y -i 6 --- (y- _l + -i
376 Y ABBAS AND J J LIANG y Theefoe, o udy he oluio of (21), i i ufficie o udy he oluio of y LEMMA 23 x i - oluio of equio (21) If 0 i iiive elee i GF(), he he followig ee e ue: -i () 0 -I Z -I k * 1 (b) Fo evey Z hee exi iege k uch h 0 whee 0 < k < -i If i, k # 0 LEMMA 24 Fo evey uch h divide d i i i iege, k ROOF Le k So, l(od k) d -i -i -i, he I / 1 k-2-1 (o) -I (o)k- + (o) + I Sice -= l(od k), we hve J l(od k) fo j 1,2 Theefoe, k-i ()j 0(od k) Hece _ i j=o 1 I LEMMA 25 If / i, 1 d ROOF Bu, fo Le 24 -! -k / - i 0 k _ -i -i -i -i i i / 1 _(/) k / I k -1 he -i i iege -i -i -l_k i iege Alo, 0 - i ilie i Theefoe, -I k 0 od(-l) So -i k -i THEOREM 21 Give x x (22) whee, i d 0() he () Equio (22) h oluio i GF() d Z O (b) If i oluio of (2 2) whee O i iiive i GF(), he k 1 (od -!) -1 -I -I (c) x 0 i he oly oluio of (22) i Z ROOF () If x Z he x x x Theefoe x O (b) Sice x x d # i, he x _GF() (c) If O i oluio of (22) he (o) -I -i k == 8 -i Thu, i (- i) =- k (od I) -i which ilie k _ 1) 1 (od -i -i -i I follow fo Le 25 h he bove coguece i eigful
(22) THEOREM 22 FUNDAMENTAL SETS OVER A FINITE FIELD 377 Fo evey k i k -i _ i) (od O i oluio of I -I -i ROOF Le -i + j j 0,1,2, he -I -i -I _l _ i -k (S) o -I (8-l)j I So, (O) O Now coide he oluio of x x + 6 (23) whee # i, Z 6 Z, l d 0() Fo Le 22, Theoe 21 d Theoe 22, we hve O + - 6 i oluio of (23), whee k -I (od d i he oly oluio i Z -i -i "-i THEOREM 23 Le A be fudel e i GF() wih idex d coefficie d i If A i o icluded i y oe ubfield of GF(), he O() l(- i) ROOF Le 0() d d u A I follow dl d h idex wih coefficie Thu, + fo oe 6 6 Z Hece d d + (ed-1 + d-2 + + 1)6 Theefoe 6GF(di), which ilie GF(d) GF() d d COROLLARY 21 Le X X + 6 (24) uch h i, 6 Z l, 1 d O(), he equio (24) h oluio i GF() d ll he oluio e icluded i GF() ROOF GF() i ubfield of GF() Equio (24) ifie he codiio of Theoe 21 ove GF() d if u i oluio, he u -1-2 GF() u + ( + + + 1)6 + O Theefoe -1 Noe: If 8 i iiive elee i GF(), he y 0 -i i iiive elee i GF() Theefoe, he oluio of equio (24) e of he fo: k y + whee 1 (od -1 0-1 -1 1 1 k d 7 1 Sice -1 -I k -1 -i -I -i -I,we hve 8 (8 O -I k -I k -l k So, k k Theefoe, he oluio of (24) ove GF() e of he fo 8 + i_-, whee k!) -I -1 -l- (od -i -i -i
_ 378 Y ABBAS AND J J LIANG LEMMA 26 Equio (24) h oluio i GF() ROOF Coolly 21 iue h equio (24) h oluio i GF() -I -i Fo he eviou oe, Theoe 21 d Theoe 22 we hve 8-1 0 d 0 + i oluio d if H {0 B + lgf()} he i oluio of (24) I i cle h H h elee O()[ THEOREM 24 Fo evey divide d evey E Z 1 uch h hee exi fudel e i GF() wih idex d coefficie ROOF By Le 26, x x h oluio i GF() If e i _ -1 ok iiive elee i 6F() he 0-1 i oluio We cli h h idex wih coefficie To ove hi, we ue h idex wih coefficie b, heefoe by Theoe 14 d Theoe 22 we coclude h i oluio of x bx d i i of he fo e whee So, hee exi j whee j > 0 uch h: -1-1 -1-1 -1 _ od 1) -I -I -I -i k he 0 _< j o_ ll_ 1 Sice _ i_ > k < - 1 d i _< we will hve -i d hi i codicio 0 <_ J < _-- Hece A--A i fudel e wih idex d coefficie COROLLARY 22 The iiu ubfield which coi ll he oluio of equio (24) i GF() ROOF The oof i diec licio of Theoe 13 d Theoe 24 he We will ow eview oe kow fc bou he oluio of x x + b i field GF() Le x x + b, whee b EGF() (25) d d gcd (, ) d The followig heoe wee give i [i] LEMMA 27 If x 0 i oluio of (25) i GF(), he fo evey d 8 -i j i, 2,, -i, x 0 + i oluio of (25) whee 8 i iiive elee i GF()
FUNDAMENTAL SETS OVER A FINITE FIELD 379 -i ROOF THEOREM 25 THEOREM 26 d d Xo + 8 -i x + (O i _ (x0+b) + e -I Xo + e 1 + b d The ube of oluio of (25) i GF() i ehe 0 o Equio (25) h oluio i GF() if d oly if, If we ue divide d b Z i he equio x x + b he we c coclude he followig: () (b) d g- c- d(), fo evey b E Z we hve E ffio b=--b=o (26) if d oly if divide Now, we c ee he followig heoe: () Theoe 27 Equio (26) h oluio i GF() if d oly if divide (b) Theoe 28 If equio (26)h oluio i GF(), i h oluio, lo if x 0 i oluio he x 0 + 0-1 j 0,1,, -i i oluio whee 0 i iiive i GF() THEOREM 29 If (26) h oluio i GF(),he, he iiu ubfield h coi ll he oluio i GF() ROOF Sice equio (26) h oluio d divide he by Theoe 27 divide heefoe GF(o) i ubfield of GF() If i oluio of (26), he we hve + b, which ilie = + b = o = GF(o) d GF()[b#0] Le GF( Z) be he iiu ubfield which coi ll he oluio of (26) heefoe GF() C GF() d # Thi ilie [ Bu ice equio (26) h oluio i GF() d by Theoe 25 d Theoe 26, g c d(,) Hece, Theefoe, LEMMA 28 If i oluio of (26) i GF() whee h idex wih coefficie he divide d i
380 Y ABBAS AND J J LIANG ROOF Theoe 1 4 ilie SIR Le _ Aue # 1 d + 6 fo SORe E Z heefoe + -i + -2+ +i) -i -2, e + b Hece 1 d + + + 1 0 which ilie b 0, bu hl codic he codiio b 0 of equio (26) COROLLARY 23 I GF(), fo evey divide d divide hee exi (-l) elee e whee 6 A uch h A A d A GF() * ROOF Equio (26) h oluio ove GF() fo fixed b Z d hee e -i diffee vlue fo b COROLLARY 24 I GF(), if i ie, he ll oluio of (26) hve idex wih coefficie i ROOF Thi i diec coequece of Le 28 THEOREM 210 I equio (26) if divide he hee exi whee i oluio of (26) d h idex wih coefficie i ROOF Sice he (26) h oluio Alo by Le 28, if h idex wih coefficie 1 whee ir, he iie x x + i (27) whee 6 --_ (b), hece l d By Theoe 27 d Theoe 28 equio (27) h oluio Le {Sl, 2, } be he e of ll he idice of he oluio of (26) uch h 1 _< < i j < R fo evey I < j Sice _< whee i he gee iege le h o equl o we hve Sl 2 < + + + < + 2 + -1 Bu k+l k+l ice < 2k fo k > i d < -I -I 2k + i we hve d hi i codicio R < -i COROLLARY 25 I GF(), fo evey divide, d divide hee exi fudel e wih idex d coefficie I 3 THE TOTAL NUMBER OF FUNDAMENTAL CLASSES IN SOME FINITE FIELDS I hi ecio we will iveige he ol ube of fudel cle fo oe ecil fiie field Fo he eviou udy we coclude h If hee exi fudel e wih idex d coefficie 1 i GF() * I GF(), fo evey divide, d evey Z # 1 whee 1 hee exi fudel e wih idex R d coefficie Sice 1 he he -l) # i g c d( If A i fudel e wih idex he A h (- l) elee
FUNDAMENTAL SETS OVER A FINITE FIELD 381 I hi ecio we will ue he followig oio: () 0() Nube of fudel cle i GF() (b) A() Nube of fudel cle i GF() bu o i y oe ubfield of GF() (c) %(,,) Nube of elee i GF() wih idex d coefficie d oe of hee elee belogig o y oe ubfield of GF() (d) E(,) {x x + Z Z d 0 if 1} " (e) SE(,,) i he e of ll oluio of he equio of E(,) i GF() bu o i We hll iveige i he followig he ube of fudel cle i GF( q ), whee q -l, d q @ +l q [q-l] LEMMA 31 If q- l,he q divide i fo evey 0,i,2, ROOF We will ove hi 1e by iducio By Fe Theoe he le, i ue fo O Aue i i ue fo he q [q-l]_ i (qs[q_l]) 1 [ q-i q-2 (q [q-l] 1) (q [q-l]) + (q [q-l]) + + [q [-ll, 11 (q [q-1]) ]--1 q Sice q [q-l] i ood q, he (q [q-l]) E i (od q) fo eve J 0,I q So, (q [q-l]) j=l +l [q_l] [q 1] S +2 q (od q) -= 0 (od q) which ilie q divide +l LEMMA 32 I GF( q whee q-1 d q #, we hve A(+l) q [q [q-l]_l] +l (-l)q, ROOF Sice g c d(q,-l) l,he fo evey 6 Z, whee @ I, q I fo evey O, heefoe equio (26) d equio (27) hve o oluio i +l +l h GF( q fo evey q h > 0 Hece evey elee i GF( q bu o i +l +l GF( q h dex q which lie h evey fudel e i GF( q bu o +l i GF( q lo h idex q So: +l A(+l q _q q [_q_ [q-if_ 11 +l +l (-l)q (-l)q Fo Le 31 d g c d(-l,q) 1 we hve A( +l) i iege COROLLARY 31 I GF( q) whee g-l, q we hve: I] O(q) i +------= i + (- l)q (q-l_ I) (-l)q (31)
382 Y ABBAS AND J J LIANG +l +l Sice O( q O( q + A( q we coclude h: -1 +l O( q 1 + A( q --O -i q i + [q [q-l] =0 (-l)q+l whee q-i, q # We hll ow udy he ube of fudel cle i GF( ) By Theoe 27, Theoe 28 d Theoe 29, he equio, whee b Z d < h +l GF( ) oluio d ll oluio e icluded i LEMMA 33 All he oluio of (32) hve idex ROOF Le be oluio of (32) d h idex We hve k fo oe k x x + b (32) k < d + c fo oe c o oluio of (32) +l COROLLARY 32 I GF( bu o i GF( ), hee e (-l) elee +l wih idex So x( I) (-l), ROOF Fo fixed b E Z equio (32) h oluio d we hve (-l), elee i Z +l +l COROLLARY 33 If e E GF( bu GF( ) he h idex o whee > I COROLLARY 34 Fo > i; A ( +l) (_l) + (-l) (_l) +l +l (-l) (-l) +l + --i (-I)-I-I --I [ (-I)-I_I (-l) + i (33) ] I GF(), he equio x x + b, (34) whee b 6 Z h oluio d ech oluio h idex Theefoe hee e (-l) elee i GF()-z wlh idex Thi ilie: LEMMA 34 O() i + + -(-l)-_ (-l) (-l) (35)
FUNDAMENTAL SETS OVER A FINITE FIELD 383 COROLLARY 35 -i +l O( 2 + + A( (36) l I wh follow we will udy he ube of fudel cle i GF( "q whee q-i, q d, > I Sice q-i he fo eve divide -q d eve E Z wih i, we "q hve i So, Theoe 13 ilie h SE(,, q) fo evey # 1 d evey l q By Theoe 27, Theoe 28 d Theoe 29, we coclude he followig: LEMMA 35 Fo evey d h; 0 < < -i, 0 < h <, SE(qh, I, Sq) h h (-l) "q elee, ll of he e coied i he iiu ubfield +l h h GF( q d SE(qh, i, Sq)N GF( "q LEMMA 36 Fo evey, h uch h 0 < < -i, 0 < h < -i, SE(qh 1 Sq4) i icluded i SE(qh+l 1 Sq) qh ROOF (qh, Sq), Le u q SE I, The u + 6 fo oe q Z h which ilie u "q + I u + [q ], whee [x] i he le oegive eidue of x od Sice [q- ] 0 he SE( h+l q i, Sq) I i cle h if u SE( q h, 1, Sq),he u h idex q fo oe 0 < < h Sice, SE(, i, Sq A) C SE( q, i Sq) C C SE (qh+l, i Sq) we coclude h +l h+l h+l h+l h X( "q q I) (-l) "q +l h+l Theefoe, i GF( q hee e hi qh (_l) "q [q-l] h h _ h h (_l) "q [ "q [q-l]_l ]= "q [ (-I) q "q q [q-l] h+l +l h+l h+l fudel cle wih idex q Fo Le 31 d he fc h + i, he ube give i (38) i iege +l h+l h+l Sice y oe ubfield of GF( "q i ubfield of GF( "q o +l h +l h+l GF( "q ) So, i GF( "q ), he ube of fudel cle wih +l h+l idex "q i equl 6 q _ q _ q + q _(_l) q [-q [q-l] +l h+l h+l +l h h h h +l h+l (-l) q +l h +l h h h Theefoe, we hve he followig q [ q [q-l]_l]_ -q [-q [q-l] h+l (-l) +l q -1]- I: (37) (38) (39)
384 Y ABBAS AND J J LIANG THEOREM 311 I GF( "q ), Fo i follow h +l ( "q +l (38) + (39) +l h+l +l h+l +l h O( "q ( "q + O( "q h+l h + O( "q O( "q (310) We ow udy he geel ce, ie he ube of fudel cle i GF() Fi, le i 2 ql q2 q uch h i # 0 gi-i d qi # fo evey i 1,2 of Le N(,) be he ube of elee i GF( ) bu o i y oe ubfield GF (S) THEOREM 312 The ube N(,) (i)q j whee i he Mblu fucio (See [2]) i-j=, Sice fo evey divide d evey E Z he e SE(,,) I ilie h if EGF (k) whee GF (k) i ubfield of GF() d doe belog o y oe ubfield of GF(k), he = h idex k So, we hve he followig THEOREM 313 If GF(k) C GF() he A(k) N(k,) k (-l) k i 2 Now le ql q2 q uch h k # O, qi # " qi2-i fo evey i 1,2,, # 0 d i l LEMMA 3 7 Fo evey 2 ql q2 q whee 0 < < k-l, 0 _< _< i i i 1,2,, he SE(,l,) h (-l) elee, ll of he e coied i he iiu ubfield GF(o) d SE(,l,) N GF() Sice SE(,l,)C GF(), we hve SE(,l,) SE(,l,) LEMMA 38 Fo evey 1 "ql q whee 0 < < k-i, 0 < < i i fo i 1,2, d fo oe < -i, SE(,l,) i oe ube of SE(q,l,) LEMMA 39 If k divide, d divide he SE(,l,) COROLLARY 36 If SE(ql,l,) # d SE(q2,1;) # fo oe I # 2 he SE,l,) # (ql SE(q 1 ) 2 The oof i iedie licio of Le 37 COROLLARY 37 If divide bu o, he SE(,l,) f SE(,l,) +l The oof i diec licio of Le 37 COROLLARY 38 If fo oe 0 < < k-i l d I bu +l 2 d he whee (e,)= g- c d(,) SE(,l,) SE(,l,) SE((,),l,) The oof i diec licio of Le 37, Coolly 36 d
FUNDAMENTAL SETS OVER A FINITE FIELD 385 Coolly 38 We kow fo Theoe 210 if divide d SE(,l,) @ he hee i SE(,l,) uch h h idex We w o fid he ube of elee i ll 12 lh SE(,l,) wih idex Le ql I "q12 q d divide whee 0 _< _< k- i, I # 0 fo l,,h d h < If we ege he q i he fcoizio of uch h ql q fo 1,2,,h, he SE(,1,) h fo evey i 1,2,,h, d i=in SE(--" 1,)qi, SE(ql q2 qh l, ) We hll ue he followig oio fo he eiig of hi ecio: i 2 h "ql q2 qh 1 < d h h I qi 2 i=l qi = qil qi2 h i<j qiqj 1 < i I i 2 < i h d qi q + fo i j=l j j =l l-i 2-1 h-1 If > h he we defie O Theefoe h "ql "q2 qh Fo eviou Le, we coclude he followig If R[SE(,I,)] i he ube of elee i SE(,l,) wih idex he R[SE(,I,)] I + 2 3 + + (-l) + + (-i) h h Alo, if B GF() d o i y oe ubfield of GF(), he B h idex o Hece we hve he followig THEOREM 3]4 &() R[S(,l,)] (,)-R[S(,l,)] (-l) (-l) (311) (3 12) To deeie he ube of equivlece cle i GF( q whee ql-l, we eed o udy SE(,,q fo ll dividig q d Z * Fo Theoe 27 we hve SE(M,l,q ) fo evey lq Alo we kow h fo evey lq d evey # I,, S/ Z if q i he SE(, qs) h (-l) elee Oe queio we will y o we fi i h fo give, i hee Z # I q/ uch h I d he how y uch i Z c oe fid? Aohe queio i fo fixed d, how y, e hee uch h SE(,,q E(,,q ) LEMMA 310 I Z if qv divide (-l) he hee e q (q-l) elee of v ode q ROOF Le b be iiive elee i Z Aue fo oe k; k I, b k i oluio x q i Thi ilie k-q 0 od(-l) So, k - -!_ fo oe q
386 Y ABBAS AND J J LIANG 1,2,,q-l Bu lo if -1 fo evey 1,2,,q-I we hve q -(b) q b -I" I d b # 1 which ilie h i Z we hve (q-l) elee of ode q Le i q h whee he g c d(-l,h) I, he fo evey uch h 0 od(q -v h) d 0 < < qv b qv we hve i oluio of x I, which I qv ilie h i Z hee e qv- elee uch h x i d x 1 Fo v I, v 2 2 we will hve q I- (q-l) q(q-1) 2 elee of ode q The e fo v I, v we will hve q i (q-l-l) q-l(q-l) elee of ode q LEMMA 311 I Z if fo fixed b whee O(b) q d i < < hee exi of ode qv whee _> v > d ifie x qv- b (313) he hee e qv- diic elee i Z * of ode qv ifyig (3]3) ROOF Noe h equio (313) h o eeed oo If i oluio, we cli h, q+l 2q+l (qv--l) q-i e diic oluio of (313) To v iq+l v-u_l ove ou cli, fi ice O() q he fo 1,2 q e q+l qvdiic c,l, i Z Alo ( -q qv- i b b Sice v q+l) v THEOREM 3]5 If b f Z,ch h O(b) q- 0 < v <, v-d v _> v, hee e q e]l of od, d ifyig (3]3) v- v- qv-v x q-l(q-l) diic d of ode q fo which he bove equio i v-i- g c d(qu+ l,q) 1 d O() q we ve O( q fo evey v, v-i v v ROOF I Z we hve q (q-l) el,e of ode q Le c e z d 0(c) q d le c q d o d # 1 d d q i Fuheoe, he ode of d i q By Le 311, we hve q elee ifyig d fo ech d Bu hee e _q q olvble d h i excly he ol ube of elee i Z hvig ode qv q LEMMA 312 If q divide -i, he q divide -i fo evey > 0 ROOF By iducio I GF( q) whee q divide -l, he e SE(l,,q) # if d oly if O() q Sice hee e (q-l) elee of ode q d fo fixed b wih O(b) q, he e SE(I b,q) h (-l) elee, heefoe GF( q) h (-l)(q-l) (q-l) fudel (-l) cle wih idex i So we coclude: Sice -= I od q So, (314) i iege O(q) (q-l) + q--(-l)(q-l)+ I (-l) -q q (q-l) + (--q-l-l) + (-l)q q-2 +q-3 + + 1 + 1 (q-l) + (314) q i + >q- 1 + (q-l) od q 0 od q =2 V THEOREM 316 I GF( q whee q divide (-l), if O() q he fo evey q qs) q +v, whee 0 < < -v we hve SE(,, SE(,,q # d SE(q v+=l,,qv+) S(q,q
FUNDAMENTAL SETS OVER A FINITE FIELD 387 ROOF By Coolly 21 we hve,, SE,,q # Theoe 24 d Coolly 22 will ily h X q X + b; b Z h oluio wih idex q d +v-i GF(q ) By Theoe 21 he oluio e of hi equio i H { O + / GF( q d i i cle h H (D GF( q i- NOTE: Thi heoe i o ue i geel I i oible h i oe ce he iiu ubfield h coi ll he oluio of x x + h oe ubfield which coi oe of he oluio Fo Fheoe 3 coclude h i CF( q ), w]ee ql-l, SE(q,b,q ) i - ube of SE(q,,q ) if d oly if + v + v d b q, wlee qv v O(b) O() q -v Alo * v fo fixed Z wih ode q whee v > O, Theoe 315 iue he exiece of excly q elee "b" i Z uch v+l q v h 0(b) q d b Hece, fo fixed wih ode q d fixed " < -v hee e q elee b i Z uch h: SE(q-i,b,q ) SE(q,,q ) o qv v+i Z whee O( i) q d (i+l)q So, we hve SE(q- (i+l) -i ) Theefoe, if we wih 0 Z uch h 0( he hee e i+i, q SE(q,i, q fo evey i 0,1,2,-1 whee - >_ 0 d qv+ (-i) g, Le -1 q h whee g c d(h,q) i The fo evey Z # i d evey >_ i uch h O() qv i < v < E d + v < we hve he followig Le LEMMA 313 I SE(q,,qS), hee e excly (q -1)-( q -1)- g elee wih idex q d coefficie +v Fo hi le, we coclude h i GF( q hee e (q -l)-(q -i (-l) -q -i) q qv-l(q_l fudel cle v d ech cl h idex q d coefficie wih ode q Le 312 iue h (315) h iege vlue We will ue he oio: -I (315) F(,,v) (315), whee + v, 1 < v < d > i I i cle ow h i GF( q whee > i, we hve -i -i (q -q )- (q -i) (q-l) (-l)q (316)
388 Y ABBAS AND J J LIANG S fudel cle d ech cl h idex q wih coefficie 1 Theefoe, fo > 2, > 2 we coclude h: O(q) O(q -l) + (316) -6-1 6 + 2 F( -v v) + (-q v=l -6-1(q (-l)q (317) whee -6-1 i{-l, -l} d 6 x{o,-} If 1 he we hve: -i -I (q -i) (q-l) 0(q ) 0(q + (3 16) + -i (-1)q REFERENCES i LIANG, Joeh J, O he oluio of ioil equio ove Fiie Field Bull Clcu Mh Soc 70(1978), o 6, 379-382 2 LONG, Adew F, Fcoizio of ieducible olyoil ove Fiie Field wih he ubiuio xq-x fo X Ac Aih XXV 1973, 65-80 3 BERT, AA, Fudlel Coce of Highhe lgeb hoeix Sciece Seie, The Uiveiy of Chicgo e, 1956 4 ALAHEN, JD d KNUTH, Dold E, Tble of Fiie Field Skhy, The Idi Joul of Siic, Seie A, Vol 26, Dec 1964, 4, 305-328
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